The Cause of Housing Market Fluctuations in China

The Cause of Housing Market

Fluctuations in ChinaAn Indirect Inference Perspective

Yue Gai

Supervisor: Prof. Patrick Minford

Dr Zhirong Ou

Economics Department

Cardiff University

A Thesis Submitted in Fulfilment of the Requirements for the

Degree of Doctor of Philosophy of Cardiff University

Cardiff Business School May 2019

Declaration

I hereby declare that except where specific reference is made to the work of others, the

contents of this dissertation are original and have not been submitted in whole or in part

for consideration for any other degree or qualification in this, or any other university. This

dissertation is my own work and contains nothing which is the outcome of work done in

collaboration with others, except as specified in the text and Acknowledgements. This

dissertation contains fewer than 65,000 words including appendices, bibliography, footnotes,

tables and equations and has fewer than 150 figures.

Yue Gai

May 2019

Acknowledgements

This thesis becomes a reality with the kind support and help of many individuals. I would

like to extend my sincere thanks to all of them.

First and foremost, I would like to express my sincere gratitude to my supervisor Prof.

Patrick Minford for the continuous support of my PhD study and research, for his patience,

motivation, enthusiasm and immense knowledge. Also, thank him for providing me with a

full PhD scholarship. I would also like to thank my secondary supervisor, Dr Zhirong Ou for

his encouragement, insightful comments and hard questions.

I thank all my fellow in office B48 for the stimulating discussion, for the days we were

working together before exam and deadlines, and for all the fun we have had in the last five

years. ’The big family’ belong to us. And also thank my friends who support me and help

me in every way.

Last, I would like to dedicate my work to my beloved parents Renlong Gai and Juan

Li for their love, endless support, encouragement and always standing by me. I would also

thank my husband Qiupeng Kong for always believing in me, encourage me and raising me

up when I was feeling down.

As a final word, I would like to thank each and every individual who has been a source

of support and encouragement and helped me to achieve my goal and complete my thesis

successfully.

Abstract

This thesis addresses two main issues related to the housing market in China. It discovers:

i) the key driving forces behind the movements of housing price and the evaluation of

the model’s capacity in fitting the data. ii) try to identify whether the Chinese housing

market can be explained better by using a model with collateral constraint. The Dynamic

Stochastic General Equilibrium (DSGE) model including the housing sector and capturing

some important features of the Chinese economy is employed to explore the above questions.

Moreover, an Indirect Inference method is used to explore these issues in an empirical way.

Estimation results show that the estimated model using Indirect Inference method can explain

the data behaviour well. The estimated model shows that the capital demand shock plays a

significant major role in explaining the housing price dynamic. In terms of the second issue,

the Indirect Inference testing results show that the model with collateral constraint cannot

provide better performance in explaining the data.

Table of contents

List of figures xv

List of tables xvii

1 Introduction 1

1.1 Background and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Methodology and Findings . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Thesis Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature Review 7

2.1 The Source of Housing price Dynamics . . . . . . . . . . . . . . . . . . . 7

2.2 Collateral Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Benchmark Model 21

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.2.1 Key Features in the Model . . . . . . . . . . . . . . . . . . . . . . 24

3.2.2 The Model Setting . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.2.3 The Linearised Housing Model . . . . . . . . . . . . . . . . . . . 41

3.3 The Method of Indirect Inference . . . . . . . . . . . . . . . . . . . . . . . 45

3.3.1 Introduction of Indirect Inference . . . . . . . . . . . . . . . . . . 45

xii Table of contents

3.3.2 Indirect Inference Testing . . . . . . . . . . . . . . . . . . . . . . 47

3.3.3 Indirect Inference Estimation . . . . . . . . . . . . . . . . . . . . . 52

3.3.4 The Choice of the Auxiliary Model . . . . . . . . . . . . . . . . . 53

3.4 Data and Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.1 Description of Data . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.4.2 The Advantage of Non-stationary data . . . . . . . . . . . . . . . . 58

3.4.3 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.5 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5.2 Properties of the Model . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

4 Model with Collateral Constraint 81

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.2 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.3.4 Indirect Inference Power Test . . . . . . . . . . . . . . . . . . . . 99

4.3.5 Error Properties on Non-Stationary Data . . . . . . . . . . . . . . . 101

4.4 Standard Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.4.1 Impulse Response Functions . . . . . . . . . . . . . . . . . . . . . 104

4.4.2 Variance Decomposition . . . . . . . . . . . . . . . . . . . . . . . 109

4.4.3 Historical Decomposition . . . . . . . . . . . . . . . . . . . . . . 113

4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

Table of contents xiii

5 Conclusion 117

References 121

Appendix A Data 127

Appendix B Impulse Responses of Main Variables 133

List of figures

1.1 The housing price index . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 The growth rate of housing price . . . . . . . . . . . . . . . . . . . . . . . 3

3.1 China real GDP per capita and pre-crisis trend . . . . . . . . . . . . . . . . 26

3.2 The Principle of testing using indirect inference . . . . . . . . . . . . . . . 51

3.3 Chinese macroeconomic data: 2001Q1 to 2014Q4 . . . . . . . . . . . . . . 57

3.4 Structure Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.5 Shock Decomposition - Real Housing Price . . . . . . . . . . . . . . . . . 74

3.6 Impulse responses to Housing Demand Shock . . . . . . . . . . . . . . . . 76

3.7 Impulse responses to Monetary Policy Shock . . . . . . . . . . . . . . . . 77

3.8 Impulse responses to Technology shocks in general sector . . . . . . . . . . 78

4.1 Structure Shocks - Model with Collateral Constraint . . . . . . . . . . . . . 104

4.2 Impulse Response to a 0.1 SE Housing Demand Shock . . . . . . . . . . . 106

4.3 Impulse Response to a 0.1 standard error Monetary Policy Shock . . . . . . 107

4.4 Impulse Response to a 0.1 standard error Productivity Shock in General Sector108

4.5 Historical Decomposition of Real Housing Price . . . . . . . . . . . . . . . 114

4.6 Historical Decomposition of Real Consumption . . . . . . . . . . . . . . . 115

B.1 Government Spending Shock . . . . . . . . . . . . . . . . . . . . . . . . . 133

B.2 Preference Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

xvi List of figures

B.3 Labour Supply Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

B.4 Productivity Shock in Housing Sector . . . . . . . . . . . . . . . . . . . . 135

B.5 Labour Demand Shock in General Sector . . . . . . . . . . . . . . . . . . 135

B.6 Capital Demand Shock in General Sector . . . . . . . . . . . . . . . . . . 136

B.7 Capital Demand Shock in Housing Sector . . . . . . . . . . . . . . . . . . 136

B.8 Government Spending Shock . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.9 Preference Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

B.10 Labour Supply Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

B.11 Productivity Shock in Housing Sector . . . . . . . . . . . . . . . . . . . . 138

B.12 Labour Demand Shock in General Sector . . . . . . . . . . . . . . . . . . 139

B.13 Labour Demand Shock in Housing Sector . . . . . . . . . . . . . . . . . . 139

B.14 Capital Demand Shock in General Sector . . . . . . . . . . . . . . . . . . 140

B.15 Capital Demand Shock in Housing Sector . . . . . . . . . . . . . . . . . . 140

List of tables

3.1 Calibrated Coefficients - Benchmark Model . . . . . . . . . . . . . . . . . 61

3.2 Steady state ratios- Benchmark Model . . . . . . . . . . . . . . . . . . . . 62

3.3 Model Coefficients: 2000Q1-2014Q4 . . . . . . . . . . . . . . . . . . . . 65

3.4 Stationarity of Residual . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5 Estimated Shocks Coefficient . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.6 Variance Decomposition of aggregate variables- Benchmark Model . . . . . 72

4.1 Calibrated Coefficients - Model with Collateral Constraint . . . . . . . . . 94

4.2 Estimated Coefficients - Model with Collateral Constraint . . . . . . . . . . 98

4.3 Comparison of the Testing Results Based on II estimation . . . . . . . . . . 99

4.4 Monte Carlo Power test- 3 variables VARX(1) . . . . . . . . . . . . . . . . 100

4.5 Monte Carlo Power test- 4 variables VARX(1) . . . . . . . . . . . . . . . . 101

4.6 Stationarity of Residual and AR parameters . . . . . . . . . . . . . . . . . 103

4.7 Variance Decomposition of aggregate variables- Collateral Model . . . . . 112

A.1 Data Description and Source of Benchmark Model . . . . . . . . . . . . . 128

A.2 Data Description and Source of Model with Collateral Constraint . . . . . . 131

Chapter 1

Introduction

1.1 Background and Motivation

Background

The housing market in China has experienced extraordinary development during recent

decades. There was no housing market before 1980, with the Chinese government controlling

housing investment and construction, treating houses as welfare goods before commencing

reform. The state allocated housing to enterprises and institutions (also known as work units),

with the work units providing apartments directly to their workers as welfare goods charging

very low rent. According to Minetti and Peng (2012), ’welfare-oriented’ public housing had

some weaknesses. First, the state could not supply enough funding to take responsibility for

housing maintenance or the increase in housing supply. Second, the low rent did not ensure

good residential conditions. Gradual and persistent housing institution reforms started to be

issued from 1980 onwards. In the very beginning, individuals could get state-owned houses

at a lower price, roughly one-third of the cost of similar privately owned housing.

Full marketisation reform in the housing market started in 1998, promoting the privati-

sation of housing. The abolishment of the ’welfare-oriented’ public housing provision and

2 Introduction

adoption of a more radically ’market-oriented’ housing provision accelerated the devel-

opment of the housing market in China. Houses were treated as a commodity at prices

determined by the market after market-oriented reform. This reform lead to the Chinese

housing market boom. Figure 1.1 presents the official housing index obtained from the China

Real Estate Index System (CREIS) 1 and shows that although full marketisation reform

started in 1998, there was no significant increase until 2002. The Chinese housing market

experienced substantial growth from 2002 until the recent global financial crisis in 2007. The

housing price index jumped from 98 points at the very beginning to around 145 points in

2006 and around 190 points in 2010, an increase of 1.5 times in the former period and almost

double in the latter period. Also, Liu and Ou (2017) highlight the dramatic increase (184%)

in the price of commercial residential housing in China between 2002 and 2014.

Fig. 1.1 The housing price indexSource: cited in Wu (2015)

The dramatic rise is not the only feature of Chinese housing prices, with volatility also a

significant factor. Minetti and Peng (2012) use the log difference of seasonally adjusted real

housing price to show the growth rate of housing price between 1998 and 2011. Figure 1.2

1The National Statistics Bureau compile CREIS, which is based on a property sample from 70 cities.

1.1 Background and Motivation 3

shows the volatility of housing prices in China during that period, with the growth rate of

housing prices changing frequently, ranging from approximately -7.5% to 10%.

Fig. 1.2 The growth rate of housing priceSource: cited in Minetti and Peng (2012)

Motivation

Considerable attention is paid by not only academia but also society to the dynamic of housing

prices, arousing wide concern and discussion. Many would agree that the development of the

housing market has made an important contribution to the Chinese economy. Fluctuations

in the Chinese housing market are also a concern, especially with the collapse of the US

housing market in 2007 fresh in the memory. The following issues related to the housing

market in China also focused my attention on undertaking research in this area.

Firstly, the large volatility in housing prices in China mentioned in the last section

attracted me to understand what determines housing price dynamics in China? Accordingly,

it is necessary to develop a theoretical framework that can maximally replicate housing

market behaviour in China.

Secondly, there is growing interest in following Iacoviello type models that use housing

as collateral to study the Chinese economy. However, whether housing collateral is important

to the business cycle is an important question to investigate. I want to ascertain whether the

model assumption (housing collateral) can fit the data in China. One reason for doubting

4 Introduction

the reasonableness of the assumption concerns the propensity of consumption in China. The

reason for introducing housing collateral in developed countries is practical, with housing

often used as collateral for a large proportion of borrowing (Iacoviello (2005)). Compared

to the high propensity to consume in developed countries, China has the highest saving

rate in the world according to Kraay (2000). Household in China usually use their saving

to consume, not borrowing, especially using housing as collateral. Baldacci et al. (2010)

highlight this, showing that two components lead to a decline in the household consumption

ratio, one being changes in the savings rate and the other the share of household income in

GDP. Accordingly, Chapter 4 will test whether housing collateral is statistically important to

the business cycle in China.

In summary, this thesis will examine two main issues: the key driving forces behind

housing price movements and the evaluation of the benchmark model’s capacity to fit the

data; ii) to identify whether the Chinese housing market can be explained better by using a

model with collateral constraint rather than the benchmark model.

1.2 Methodology and Findings

Methodology

A Dynamic Stochastic General Equilibrium (DSGE) model with Indirect Inference evaluation

and estimation are employed to explore the above questions. In particular, some important

features of the Chinese housing sector are considered in my model. Firstly, two sectors are

allowed on the supply side of the economy with explicit modelling of the price and quantity

of the housing sector to study the behaviour of the housing sector. Secondly, productivity

shock in both housing and general sectors are assumed to be non-stationary. The reasons for

including these features in the model will be carefully discussed in Chapter 3.

1.2 Methodology and Findings 5

There are two contributions in this thesis. First, the New Keynesian dynamic stochastic

general equilibrium (DSGE) model is set up by incorporating housing sector and some

important features of the Chinese economy, providing a framework to describe the Chinese

housing market in reasonable detail. Second, differing from previous literature, this research

employs a different evaluation and estimation strategy - Indirect Inference method.

In order to check whether a theoretical framework can explain housing market behaviour

in China, a powerful Indirect Inference testing procedure is employed to apply in the New

Keynesian DSGE model with the housing sector. The Indirect Inference method evaluates

the model’s capacity to fit the data by providing a classical statistical inferential framework,

as introduced by Minford et al. (2009) with Le et al. (2011) refining this method using Monte

Carlo experiments. The evaluation aims to compare the simulated data generated by the

model and the actual data through the auxiliary model . A cointegrated vector autoregressive

with exogenous variables (VARX) has been chosen as the auxiliary model. The Wald statistic

is employed as the criterion for evaluating the model, which compare the Wald statistic

calculating using simulated data with using actual data.

The Indirect Inference estimation strategy is also used in this thesis. This estimation

method is widely used in the estimation of structural models, such as by Smith (1993),

Gregory and Smith (1991), Gourieroux et al. (1993) and Canova (2007). The idea behind the

Indirect Inference estimation is to search for a set of parameters that are best able to satisfy

the test criterion, with the details of Indirect Inference testing and estimation procedure

introduced in Chapter 3.

Findings

The main empirical findings related to the above two research questions reveal: i) Indirect

Inference testing results show that the data reject the model using the calibration values.

However, the estimated model using Indirect Inference method can explain the data behaviour

6 Introduction

well. I discover the housing market using the estimated model. Concerning the driving force

behind fluctuations in the Chinese housing market, the variance and shock decomposition

suggest that capital demand shock plays a significant major role in explaining housing prices.

ii) Indirect Inference testing results show that the model with collateral constraint does not

perform better at explaining the data. The benchmark model using the Wald statistic as a

guide is the best model.

1.3 Thesis Structure

The remainder of this thesis is structured as follows. Chapter 2, following the above two

motivations, summarises literature on the volatility of housing prices in term of theoretical

and empirical works, and reviews structured DSGE models with collateral constraint and the

transmission mechanism working behind it. Chapter 3 focuses on exploring the first research

question concerning the key driving forces behind movements in the housing sector. In order

to answer this question, a New Keynesian DSGE model incorporating the housing sector

and some important features of the Chinese economy has been established as the benchmark

model. This theoretical framework is then evaluated using Indirect Inference testing and

estimated during the sample period by using Indirect Inference estimation. Standard analyses

of housing price dynamics are also presented. Chapter 4 focuses on the second research

question. One more feature, collateral constraint, is added to the benchmark model to identify

whether the Chinese housing market can be explained better using a model with collateral

constraint than the benchmark model. The Indirect Inference method is used to discriminate

between these two models, with Monte Carlo experiments showing how powerful the test is.

In addition, empirical analyses of the collateral model are displayed at the end of this chapter.

Chapter 5 concludes all the findings of the different chapters.

Chapter 2

Literature Review

As mentioned in Chapter 1, there are two main research questions relating to the Chinese

housing market I am going to answer in this thesis: i) The key driving forces behind the

movements of housing price and the evaluation of the model’s capacity in fitting the data.

ii) try to identify whether the Chinese housing market can be explained better by using a

model with collateral constraint compared to the benchmark model. Following these two

motivations, this chapter surveys the literature on the housing market. More specifically,

Section 2.1 summarises the literature on the volatility of housing price in terms of theoretical

and empirical works. Section 2.2 reviews the structured DSGE models with collateral

constraint and the transmission mechanism working behind it.

2.1 The Source of Housing price Dynamics

Empirical

In the existing empirical literature, the housing price fluctuation is affected by the economic

fundamentals. The main fundamental explanatory factors are construction costs, disposal

8 Literature Review

income and population. There is no consensus among researchers regarding the source of

housing price dynamics in the existing empirical literature.

Case and Shiller (1990) find various fundamental factors can explain the variation in

housing prices, especially positively correlated with the change in construction costs, pop-

ulation growth and disposal income. The analogous results given by Clapp and Giaccotto

(1994) show that population growth and employment have considerable forecasting ability

to forecast the residential housing price variations. Capozza et al. (2002) use panel data to

explore the driving force of real house price dynamics. The results of their research show that

shocks such as growth rates,and construction costs affect house price differently. The high

real income growth and high real construction costs lead to real house price continue to rise,

which cause significant overshooting. However, population growth does not have explanatory

power in explaining real house price dynamics. The results coincide with the findings given

by Poterba et al. (1991). He attempts to explain why housing prices vary so dramatically in

the US using the regression model. The empirical results suggest that household income and

construction costs are the most important driving force leading house price dynamics.

Potepan (1996) include more social environmental variables into his research such as

rent, land prices, household income, population, quality of public services, criminal rate, air

pollution, inflation, mortgage, interest rates, property tax rate, construction costs, agricultural

land prices and legal land use constraints. The results show that household disposable income

and construction costs have stronger explanatory power on house price fluctuation.

Some fundamentals variables considerably influence housing price in the short-term,

other variables have more explanatory power in the long-term. Quigley (2002) employ 41 U.S.

metropolitan areas data over a fifteen-year period to study the average housing price variation

influenced by economic fundamentals. The empirical findings show that some fundamentals

variables such as unemployment rate, housing supply and construction permission cannot

2.1 The Source of Housing price Dynamics 9

give the powerful explanation in housing variation in the short run, but explain well in the

long run.

Another explanation for the fluctuation of housing price is monetary policy. Jud and

Winkler (2002) study the dynamics of real housing price appreciation in 130 metropolitan

areas across the United States. Their study finds that not only population growth, real

income changes can strongly affect real house price, but monetary policies also influence

the variation in housing price in the long run. Ahearne et al. (2005) focus on the study

of the influence of monetary policy on the housing price dynamics. The empirical results

show that monetary policy plays a significant role in explaining the fluctuation of the house

price. Similar results reported by Jacobsen and Naug (2005) show that interest rates, housing

construction, unemployment rates and household income play an important role in explaining

the house price dynamics in Norwegian.

For China, researchers also want to explore whether these factors have the same ex-

planatory power on house price dynamics. They explore Chinese housing market influenced

by the fundamental factors from both demand and supply sides. Many have the similar

conclusion that the fluctuation of housing price in China is mainly a reflection of the market

fundamentals.

Li and Chand (2013) study the contribution of market fundamentals to house prices in

urban China using annual data from 29 provinces. Their findings show that the level of

income, construction cost and user cost of capital are the primary determinants of house

prices. It is quite interesting to find that the supply factors including construction costs, the

user cost of capital play a significant role in explaining more developed provinces. Wang

and Zhang (2014) evaluate the importance of fundamental changes in explaining the rising

housing prices in China. The results suggest that the fundamental factors such as population,

wage income and construction costs can account for a major proportion of the housing price

rise. Similar results can be found in Chow and Niu (2015). They use annual data to show

10 Literature Review

that the fundamental economic factors in both demand and supply side can explain well in

the variation in housing prices, with income determining demand and construction affecting

supply. Deng et al. (2009) agree the conclusion that the fundamental factors such as income,

housing supply and construction cost are the important determinant, but in their research,

interest rate and population growth cannot explain the variation of housing price.

Monetary policy is also an explanation for the house price dynamics in China. Some

researchers believe that the monetary policy plays a significant major role in explaining

the real housing price in China rather than economic fundamentals. Xu and Chen (2012)

employ quarterly data from 1998 to 2009 to study the impact of monetary policy variables

on the fluctuation of house prices in China. Empirical results suggest that the volatility of

housing prices is mainly driven by monetary policy, which an expansionary monetary policy

increase the growth of housing price while restrictive monetary policy decreases the growth

of housing price. The similar results can be found in Zhang et al. (2012), Yu (2010) and Guo

and Li (2011). They believe that monetary factors such as bank loan rate, excess liquidity,

money supply growth, mortgage rate and mortgage down payment requirement can explain

the housing price dynamics well.

There is no consensus among researchers regarding the source of housing price dynamics

in the existing empirical literature using a single regression model. Liu and Ou (2017) give

the explanation why use single regression model may come across such an ambiguity. The

first reason they summarised is using a single regression model may exit the omitted variable

problem when the ’equilibrium conditions’ are derived and put forward for estimation. Hence,

it is easy to understand why some factor is shown to be significant in one model, but in other

models not. It may be because the model has failed to consider other important factors that

would reflect the facts.

The second issue when using this method is endogeneity problem, which forces econome-

tricians either to assume these variables are exogenous such as Deng et al. (2009) just cited,


or using ’instruments’ to avoid inconsistent estimation. Liu and Ou (2017) list two reasons

showing that endogeneity problem does not go away even employing a more inclusive model,

which is just inherent in any model version where equilibrium is estimated with a single

equation. On the one hand, the economic interactions as reflected by the data would be

artificially abandoned in the modelling process by imposing exogeneity. On the other, the

partial equilibrium model omits the information about the rest of the world. The endogeneity

arises due to little information about the ’true’ instruments, which can overstate the standard

error of the coefficients of these variables causing some variables to be shown insignificant

even they are important.

These studies of the housing price dynamics using the various econometric models exist

the above two issues that cannot solve to its root. Therefore, some researchers go one further

to employ a dynamic econometric model (VAR or VECM). There are some advantages

of using VAR and VECM model. On the one hand, VAR and VECM can circumvent the

endogeneity problem through using lag for all explanatory variables. On the other hand,

some factors such as gender, marriage and urbanisation are difficult to model in a structural

model, but VAR and VECM can consider as one of the explanatory variables. (Liu and Ou

(2017))

Vargas-Silva (2008) study the importance of monetary policy shock in explaining the

housing market in the U.S. using VAR. The results show that monetary policy shock plays

a significant role in explaining the house price dynamics. There is a negative relationship

between housing price and contractionary monetary policy shock. Lastrapes et al. (2002) use

a different identifying restriction to study the impact of money on the housing price. They

have a similar conclusion that money supply shock contributes significantly to the variance

in housing price. Gete (2009) use an SVAR to study the housing market in OECD countries.

He finds that housing demand shock is the essential factor for house price dynamics.


In terms of the literature of housing price dynamics in China using VAR or VECM, Bian

and Gete (2015) employ VAR identified with theory-consistent sign restrictions to study

housing dynamics in China. They consider seven potential determining factors such as

population increase, credit constraint, housing preference, savings rate, tax policy, change in

land supply and productivity progress. Their results suggest that productivity, savings and

policy stimulus play an important role in explaining the housing price dynamics in China,

even if all shocks play relevant roles. Garriga et al. (2017) study the importance of the

structural transformation and urbanisation process to the Chinese housing market. Their

findings suggest that supply factors and productivity are the dominant drivers in housing

price dynamics in China.

However, according to Liu and Ou (2017), there are some limitations that VAR or VECM

cannot address. In terms of policy analyses, there is little information about the transmission

mechanism that policymakers would be interested since these reduced form models cannot

provide such information about how the housing price is determined. Although some

researchers try to use theoretical restrictions on estimating to cover this issue, however, the

implication is often sensitive to the imposed restrictions. Therefore, a micro-foundation

structural model is chosen in this thesis to study the housing market dynamics in China,

which can show the causalities among economic variables that established as a result of

different agents’ interactions with their optimal choice. Hence, it is necessary to set up a

model that can capture the transmission mechanism and fit the data well.

Theoretical

A micro-founded dynamic stochastic general equilibrium (DSGE) model is widely used to

study the dynamics of the housing market and the transmission mechanism working behind

it. The increasing researchers have followed Iacoviello (2005) and Iacoviello and Neri (2010)

to discover the housing market fluctuation, which use housing as collateral for loans to study


the housing sector and business cycles. In their extended model, the collateral constraint is

faced by both firms and impatient households. Iacoviello and Neri (2010) construct a DSGE

model including a rich housing sector to a framework to study the sources and consequences

of fluctuations in the U.S. housing market using the Bayesian method. On the supply side

of their model, they consider a multi-sector structure with different rates of technological

progress to capture some important observations in the housing market. The other feature

of their model is on the demand side. They introduce the collateral constraint by splitting

households into two different types: patient (lenders) and impatient (borrowers). They treat

the constraint as a channel to emphasise the spillovers effect, which the increase of housing

price affect borrowing and consumption of constrained households. Their results show that

housing demand shock and productivity shock in the housing sector are the main driving

force of the volatility of housing prices. The contribution of monetary factors in housing

price appears more important in the long term. In terms of consequences of fluctuations,

their results show that the collateral constraint amplifies the effects on consumption given the

increase of housing price.

There is growing interest in Iacoviello-type model studying the driving forces of housing

price dynamics in China. More factors are considered to enrich the model based on their

analysis framework in the following literature. Minetti and Peng (2012) focus on the demand

side and try to identify whether there is social psychology - the ’keeping up with the Zhangs’

behaviour - that influence households’ behaviour and thus drives the fluctuation of housing

price. They include the factor of ’keeping up with the Zhangs’ in the utility function and

assume that there is a positive relationship between the household’s utility and individual

consumption in housing purchases. On the contrary, there is a negative relationship between

the household’s utility and society’s average consumption in housing services. Their Bayesian

estimation results show that there is ’keeping up with the Zhangs’ and the presence of this

social psychology play a significant major role in explaining the volatility of real housing


prices. Liu and Ou (2017) focus on the banking system to investigate the source of housing

price dynamics, which allow for a ’shadow’ bank affiliated to the ’normal’ bank capturing

Chinese economy. They find that housing demand shock is the main driving force for housing

price fluctuation, which accounts for over 80%.

In terms of the DSGE framework, monetary policy variable is also an important explana-

tion for the fluctuation of the house price. Researchers analyse different monetary policies to

study how to stabilise the housing market in China. Ng (2015) use an estimated DSGE model

with a Taylor rule to discover the sources and consequence of fluctuations in the Chinese

housing market. In addition, they also discover what is housing demand shock in China.

Their model is based on Iacoviello and Neri (2010)’s framework with sectoral heterogeneity

on the supply side and collateral constraint considered on the demand side. Their estimated

results show that housing demand shock is the main driving force in explaining the house

price dynamics. Monetary policy also contributes significantly and appears more important

in the 1990s. Ng (2015) employ a price rule - Taylor rule to study, while Wen and He (2015)

adopt a quantity rule - McCallum rule to discover the key driving force of housing price

fluctuations in China. Money supply and credit constraint are considered in their model to

capture some features of the Chinese economy. Empirical results show that housing demand

shock plays an important role in explaining the fluctuation of the house price. Money supply

shock cannot explain housing price movements compared with housing demand shock. On

the other hand, their policy suggestion shows that it is better to include the real housing price

in monetary policymaking. The combination of real housing price and money supply rule

can stabilise the Chinese economy. Zhou et al. (2013) consider in a similar vein. In order to

study how to stabilise the expanding housing market, they summarised a series policies that

the Chinese government issued into four different categories: land policy, monetary policy,

property tax policy and affordable housing policy. The empirical results show that a policy

2.2 Collateral Constraint 15

mix can keep the housing market stable, which the property tax policy control the demand

side while the land policy adjusts the supply side.

In summary, most of the literature that using a micro-founded DSGE model employing

Bayesian estimation have come to conclude that the housing demand shock plays an important

role in explaining the fluctuation of housing price in China. Policy suggestion given by

the above literature shows that some policy such as property tax and property purchasing

limitations could affect housing demand directly so that to decrease the house price and keep

the housing market stable.

2.2 Collateral Constraint

In the last section, I summarise the literature about the sources of fluctuations in the Chinese

housing market. In this section, I am going to focus on the literature about collateral constraint

in the structured DSGE model. We learn a lesson from some developed countries that a

slump in housing prices might have a seriously negative effect on the wider macroeconomy.

The reason is the housing property is usually used as a significant collateral. Therefore, the

transmission mechanisms is set up through the collateral constraint and link the housing

market and the real economy.

The model with collateral constraint

I introduce a channel that connects the housing market and the wider economy: the collateral

constraint. There are different ways to introduce the collateral constraint into the structure

model either on the firm side or the household side. In this thesis, I focus on the household

side, which follows Iacoviello and Neri (2010) and includes the collateral constraints into the

structured DSGE model.


The increasing interest in DSGE housing model literature have focused on the role of

collateral constraint.The collateral constraint is first introduced to explain the financial crisis

by Kiyotaki and Moore (1997). The line of this research introduces how collateral constraint

interact with aggregate economic activity over the business cycle. More specifically, they

endogenise the collateral constraint that limits the borrowing capacity. There are two types

of agents in their framework: patient agent and impatient agent. The patient agents are

called gatherers in their paper, which is a saver. The impatient one are called farmers in

their paper, which can be thought as entrepreneurs or firms that wish to borrow from the

patient agent to finance their investment projects. The difference between the patient agent

and impatient agent is that they have a different rate of time preference. The collateral

constraint is faced only by the impatient agent.1 Therefore, loans will only be made when the

impatient household use some other form of capital (such as land, buildings and machinery)

as collateral. The borrowers’ credit limit and an investment decision are affected by the value

of the collateral asset and the tightness of the credit market. That implies if the value of

durable assets decreases for any reason, the borrowing capacity of the impatient household

also decreases. In such an economy, A significant transmission mechanism is generated

through the dynamic interaction between credit constraint and asset prices, which the effects

of exogenous shocks persist, amplify and spread out.

The transmission mechanism of collateral constraint in Kiyotaki and Moore (1997) shows

that how a small scale, temporary shocks to productivity or income distribution can give rise

to large changes in production and asset prices and also their effects spillover to the rest of the

economy. The key point in their paper is the collateralisable asset plays two different roles in

their model: i) they are a factor of production. ii) they serve as collateral for loans. Suppose

that there is a negative productivity shock, which reduces the land price. The decrease in land

price reduces the net worth of the impatient agents because land is the collateralisable asset.

1The collateral constraint in Kiyotaki and Moore (1997) is: Rtbt ≤ qt+1kt , where Rt is the nominal interestrate, bt is the amount of borrowing, qt+1 is the durable asset price in the next period,kt is the durable asset.


The constrained agents are forced to reduce their investment, which also affects them in the

next period. The credit cycle works like this: less revenue they earn (due to less investment),

less net worth they gain. Again they reduce investment because of credit constraints. That

implies the temporary shock in period t has a significant impact on the behaviour of the

constrained agents not only in period t but also in following periods. There are two factors

affect the amplification of the shock: the credit limit and the price of the collateralisable

asset. Therefore, a significant transmission mechanism is generated through the dynamic

interaction between credit limits and asset prices, which amplify the shocks and spillover to

the economy.

Extension of the model with collateral constraint

Following Kiyotaki and Moore (1997)’s work, Iacoviello (2005) extend his work by including

two features. First, instead of using land as the collateral, he uses housing stock owned by the

entrepreneurs as the collateral to borrow. Second, he uses nominal debts like Christiano et al.

(2010). He studies a monetary business cycle model with endogenous collateral constraints

and nominal debt. The estimation results show that the collateral effects significantly improve

the efficiency of the economy to a positive demand shock. In particular, Iacoviello and Neri

(2010) consider the collateral constraint in the housing market. They construct a dynamic

stochastic general equilibrium model with collateral constraints estimated using Bayesian

methods to study the source and consequence in the US housing market.

There are two important features of housing captured by the DSGE model of the housing

market they developed. The first is sectoral heterogeneity on the supply side. The second is

the collateral constraint on the demand side. On the supply side of the economy, Iacoviello

and Neri (2010) allow for multiple sectors with different rates of technological progress. The

non-housing sector employs labour and capital to produces consumption, business investment

and intermediate goods. The housing sector using capital, labour land and intermediate goods


to produce new houses. In their model, following most of the DSGE literature, nominal

wage rigidity is presented in both the non-housing and housing model and price rigidity is

only allowed in the non-housing sector. The reason for developing the multi-sector structure

is based on the observation of the housing market. The post-world-war-II U.S. data show

that the relative price of housing has a long-run upward trend. The probable reason is

heterogeneous trend technological progress between the housing and other sectors of the

economy.

The second feature of their model is the collateral constraint on the demand side. Ia-

coviello and Neri (2010) introduce this constraint on the demand side by splitting households

into two different types: patient household (lenders) and impatient household (borrowers).

Similar in Kiyotaki and Moore (1997), the difference between patient and impatient house-

hold is they have a different rate of time preferences. Patient households buy consumption

goods and housing goods and also supply labour. They lend funds to both firms and impatient

household. Impatient households also buy consumption and housing goods and supply labour.

The difference is they need to borrow money from the patient household to finance their

down payment due to their high impatience. Hence, the change in housing price affects the

behaviour of the impatient household.

The collateral constraint is one of the important feature in Iacoviello and Neri (2010)’s

work. The transmission mechanism of collateral constraint in Iacoviello type model work

as following. When there is a positive demand shock, the demand for housing rise, housing

price also increases. The rise in asset prices increases the borrowing capacity of the debtors.

That implies they can borrow more due to the high asset prices, allowing them to spend

and invest more. The change in investment will cause the output to fluctuate, which in turn

influences the current asset price. Therefore, a significant transmission channel is generated

through the dynamic interaction between the credit constraint and asset prices.


Based on their analysis framework, more types of shocks and frictions are introduced

to study the housing market. Ng (2015) employ Iacoviello type model to study the sources

and consequences of the fluctuations in the Chinese housing market. In terms of the nature

of shocks driving housing price dynamic, they find that housing demand shock explains the

majority of the fluctuations in housing price. In terms of spillover effect work through the

collateral constraint, there is not a unique way to quantify the effect, which depends on the

nature of shocks. Housing demand shock has a larger contribution to the spillover effect

compared to the technology shock. However, the technology shock plays a negligible role

in the spillover effect. Liu and Ou (2017) use a DSGE model with a collateral constraint

considering shadow bank to study the Chinese housing market. Apart from investigating

the main driving force of housing market fluctuation, they also study the housing market

spillovers effect in China. They find that there is a weak spillover effect from the housing

market to the wide economy. He et al. (2017) employ a Bayesian DSGE model with collateral

constraints to investigate the interaction between the housing market and the business cycle.

They find that the collateral constraint plays a significant role in explaining the fluctuate of

the business cycle in China, which amplifies the impact of various economic shocks.

Chapter 3

Benchmark Model

3.1 Introduction

Based on the background of the housing market in China discussed in Chapter 1, we know that

the Chinese housing market has experienced extraordinary growth during the past decades.

In the very beginning, the individuals could get the state-owned houses at a meagre price

which is only one-third of the cost of housing. The full marketisation reform started in July

1998 in the following stages. The housing market in China has experienced the first round of

market boom since that. Liu and Ou (2017) mentioned in their paper, there is a considerable

increase (184%) of commercial residential housing price in China over the period between

2002 and 2014. Besides, according to Minetti and Peng (2012), the Chinese housing prices

are volatile. They show that the growth rate of housing prices approximately ranged from

-7.5% to 10% and the growth rate changes frequently. Therefore, these factors raise my

interest to think about what is the main driving force behind housing price fluctuations in

China. This is also one of the research questions I listed in Chapter 1, which I am going to

answer in this chapter.

As Reviewed in Chapter 2, the Chinese housing market has been attracting the increasing

economists to study although it does not exist for a long time. A dynamic stochastic

22 Benchmark Model

general equilibrium (DSGE) model constructed by Iacoviello and Neri (2010) estimated

using Bayesian methods is widely used to identify the main driving force of housing price

fluctuation in China and study the transmission mechanism working behind it. Ng (2015)

use an estimated DSGE model with a Taylor rule (price rules) to discover the sources and

consequence of fluctuations in the Chinese housing market. They find that not only housing

preference shock, monetary policy shock also contribute significantly to the volatility of

housing prices in China. While in the same year, Wen and He (2015) adopt another policy

rule, McCallum rule (quantity rules), to check whether it can stabilise the housing market.

They show that housing demand shock is the main driving force in housing price dynamics,

and a real house price-augmented money supply rule is a better monetary policy for China’s

economic stabilisation. Minetti and Peng (2012) using a DSGE model to analyse China’s

housing market in a different way. They focus on the demand side and try to identify whether

there is a social psychology force that affects households’ behaviour in the housing market

and thus drives the housing price dynamic. The results show that the social psychology

"keeping up with the Zhangs" plays an important role in explaining housing price dynamic.

Liu and Ou (2017) employ a DSGE model to investigate the driving force of housing price

dynamics in China. In order to capture the situation in China, they model the featured

operating of the ordinary and ’shadow’ banks in China. They have the similar findings that

the housing demand shock is the essential factor of the housing price fluctuation. In summary,

most of the literature that using a micro-founded structural DSGE model employing Bayesian

methods have come to conclude that the housing demand shock plays an important role in

explaining the fluctuation of housing price in China.

It should be noticed that none of the previous DSGE literature about Chinese housing

market evaluates the model’s capacity in fitting the data. It is quite important to evaluate

how best the empirical performance of DSGE models is. Therefore, this gap is going to

be filled in this chapter. A powerful testing procedure (Indirect Inference) is employed to

3.1 Introduction 23

apply in the New Keynesian dynamic stochastic general equilibrium (DSGE) model with the

housing sector in China and check whether this theory can explain China’s housing market.

The Indirect Inference evaluation is proposed initially in Minford et al. (2009) and refines

by Le et al. (2011) who evaluate this method using Monte Carlo experiments. This testing

aims to compare the simulated data with the actual data through the auxiliary model. An

auxiliary model that is entirely independent of the theoretical one is used in this approach to

generate a description of data against the performance of the theory. A cointegrated vector

autoregressive with exogenous variables (VARX) is chosen as the auxiliary model. The

Wald statistic is employed as the criterion for evaluating the model, which compare the

Wald statistic calculating using simulated data and using actual data. For Indirect Inference

estimation, a set of parameters that are best able to satisfy the test criterion are found when

carried out the testing. In the empirical procedures, Indirect Inference is used to test the

model on some initial parameter values that mainly based on previous literature. If the

structured model with calibrated value cannot pass the test, Indirect Inference estimation

is used to improve the overall performance of modelling fitting, which is based on Indirect

Inference testing. It allows the parameters to move flexibly to the values that maximise

the criterion of replicating the data behaviour. The detail of Indirect Inference testing and

estimation procedure are going to be introduced in Section 3.3.

This chapter is organised as follows: In Section 3.2, I first highlight some features of

the model in this chapter and then display the model setting. The principles and procedures

of the indirect inference method for evaluating and estimation are explained in Section 3.3.

Section 3.4 displays data description and also shows the calibration of the structure model.

Empirical results are discussed in Section 3.5. Firstly, I present the estimation and testing

results and then check the properties of the model like impulse response functions, shock

and variance decomposition. Section 3.6 is the conclusion part.

24 Benchmark Model

3.2 Model

3.2.1 Key Features in the Model

As mentioned in Chapter 1, there are two main research questions relating to the Chinese

housing market I am going to answer: i) the key driving forces behind the movements of

housing price and the evaluation of the model’s capacity in fitting the data. ii) try to identify

whether the Chinese housing market can be explained better by using a model with collateral

constraint compared to the benchmark model. A Dynamic Stochastic General Equilibrium

(DSGE) model with Indirect Inference evaluation and estimation are employed to explore the

above questions. This chapter focus on the first issue that what is the sources of fluctuations

in the Chinese housing market. There are some important features of the Chinese housing

sector considered in my research. First of all, two sectors are allowed on the supply side of

the economy with explicit modelling of the price and quantity of the housing sector to study

the behaviour of the housing sector. Secondly, the productivity shock in both housing and

general sector are assumed to be the non-stationary shock.

In terms of the first feature, early literature on using a micro-foundation based DSGE

modelling approach studying the housing sector usually construct a multi-sector structure

which includes housing and non-housing products in a Real Business Cycle (RBC) model

such as Campbell and Ludvigson (1998), Davis and Heathcote (2005) and Baxter (1996).

They allow homogeneity among different sector enjoying the same competitive attribute -

perfect competition. However, in the Real Business Cycle (RBC) model, money is typically

said to be neutral in both the long run and short run. Some monetary transmission mechanism

cannot work in this scenario. As we know from some previous literature, monetary policy

variables play a significant role in explaining the real housing price. I also want to check

how monetary policy works in the structure model. In Iacoviello and Neri (2010)’s work,

price rigidity is introduced in the general sector and keep the housing price flexible. There

3.2 Model 25

are several reasons why housing might have the flexible price. According to Barsky et al.

(2003), housing production is very sensitive to a monetary contraction, while the production

of general goods not. More specifically, the value of new houses decreases by almost 10%

compared to CPI when there is a monetary contraction. They also show that compared to the

inflation persistence of CPI, there do not exhibit any inflation persistence of new houses. As

in Barsky et al. (2007), a high value on the housing allows a bargaining space on the price of

housing goods. I follow their idea to construct the firm side with two sectors but simplify

their setting, which assumes factor market in both sectors operates perfectly competitive.

In terms of the second feature of the model, other than most previous literature, the

productivity shocks in both housing sector and general sector are assumed to be non-stationary.

The reason for setting non-stationary productivity shock is practical and substantial: practical

because, empirically, after the financial crisis, the output cannot go back to the previous level.

Le et al. (2014) use Figure 3.1 to show this stylized fact in China, which shows the level of

output cannot reach its previous level after the crisis. The non-stationary shocks could shed

light on the large deviations from steady time trends that economies experience no matter

booms or crises; Substantial because, non-stationary is the feature of macroeconomic data.

On the other hand, a model using nonstationary data could explain the large deviations from

steady state, which those models using stationary data do not. The business cycle model

focus on studying the dynamics and choice of macroeconomic policy on stabilising the

fluctuations, which try to abstract from the uncertainty surrounding the economy’s long-term

future and eliminate the trends from the data so as to make it stationary. Hodrick-Prescott

(HP) and Band Pass (BP) filters are the most common techniques that used in trend-removal.

However, HP and BP filters are a mathematical tool used in the business cycle to decompose

the raw data into cyclical and trend component, which are not based on theories. Hence, the

precision of the driving process that leads to trend behaviour cannot be identified using these

techniques. In addition, according to Cogley and Nason (1995) and Murray (2003), they

26 Benchmark Model

study the spurious dynamic causing from HP and BP filters to non-stationary data and show

that these filters cannot distinguish between difference-stationary and trend-stationary. The

business cycle dynamics can be generated using the HP filter even if they are not present in

the original data. Therefore, instead of using filtered data, the non-stationary data are used to

evaluate and estimate the model.

In addition, according to Le et al. (2014), they develop a model of the Chinese economy

using a DSGE framework with a banking sector based on non-stationarity to shed light on the

banking crisis in China. The model with non-stationary productivity shock can successfully

explain China’s economy well. Therefore, I follow Le et al. (2014) to propose a DSGE model

with non-stationary productivity shock to study the Chinese housing market in my research.

Fig. 3.1 China real GDP per capita and pre-crisis trendSource: cited in Le et al. (2014)

3.2 Model 27

3.2.2 The Model Setting

The households on the demand side of the economy try to maximise their lifetime utility by

choosing general consumption goods as well as bond, supplying labour and accumulating

housing in each period. There are two goods sectors on the supply side: housing goods

sector and general goods sector. Housing sector produces new housings, and general sector

produces general consumption goods. Assuming labour and capital markets in both sectors

operate perfectly competitive and factors flow freely across two sectors. Price rigidity is

allowed in the general sector and flexible price presents in the housing sector. Taylor rule

is used as monetary policy by the central bank. A various of shocks are introduced in the

economy, which will be specified in the model.

Households

There is a continuum of measure one of households. The household’s decisions consist

of maximising lifetime utility subject to a period by period budget constraint. Assuming

a constant relative risk aversion utility function (CRRA), the representative households’

lifetime utility can be written as

U = E0

∞

∑t=0

βtε

pt

[C1−σc

t

1−σc+ ε

ht

H1−σht

1−σh− ε

ltN1+η

t

1+η

](3.1)

where E0 is the expectation formed at period 0, β ∈ (0,1) is the discount factor. The

households obtain utility from general consumption goods Ct ,houses Ht and disutility from

labour supply Nt . The parameters σc,σh are the inverse of intertemporal elasticity of substi-

tution of consumption and housing, while η denotes the inverse of the elasticity of labour

supply with respect to real wage. It measures the substitution effect of a change in the wage

rate on labour supply.

28 Benchmark Model

Three shocks are introduced in the utility function: εpt ,ε

ht and ε l

t . The terms εpt and ε l

t

capture the shocks to intertemporal preferences and to labour supply. The shock εht is what

the previous literature called housing preference shock or housing demand shock. According

to Iacoviello and Neri (2010), the housing demand shock can be some social, institutional or

income changes and so on, which might shift households preferences on purchase housing

relative to other consumption goods. According to the literature, all these three shocks are

assumed followed an AR(1) process:

lnεpt = ρp lnε

pt−1 + vp,t (3.2)

lnεht = ρh lnε

ht−1 + vh,t (3.3)

lnεlt = ρl lnε

lt−1 + vl,t (3.4)

where vp,t , vh,t and vl,t are independently and identically distributed i.i.d. processes with

variances σ2p , σ2

h and σ2l .

The households’ period by period budget constraint in real terms is given by:

Ct + ph,t [Ht − (1−δh)Ht−1]+Bt = wtNt +(1+ rt−1)Bt−1 +Πt (3.5)

From equation (3.5), it should be noticed that the households can use his wealth in each

period to buy consumption goods, bond and also to accumulate houses. Note that the housing

price is the relative price. All of these outflows of funds of the households is shown on the

left-hand side of equation (3.5). The households’ wealth on the right-hand side consists of

real wages wt earned from supplying labour Nt , the interest rate gain of bond holdings from

the previous period (1+ rt−1)Bt−1 and also the real profits Πt from firms. Then, the aim of

the households is trying to maximise the utility function (3.1), subject to the budget constraint

3.2 Model 29

(3.5) by choosing Ct , Nt , Bt and Ht via the Lagrangian. Given the first order conditions, there

comes:

εpt C−σc

t = λt (3.6)

εlt ε

pt Nη

t = λtwt (3.7)

λt = βEtλt+1(1+ rt) (3.8)

λt ph,t = εpt ε

ht H−σh

t +βEtλt+1(1−δh)ph,t+1 (3.9)

For all the above equations, the marginal utility loss of choosing relevant allocations is

shown on the left-hand side. Compared to that, the right-hand side expresses the marginal

utility gain. Combining equation (3.6) and equation (3.8), we could get the well-known Euler

equation. It is a dynamic optimality condition showing a dynamic optimality decision for

consumption in the present and the future. The optimal intra-temporal substitution between

labour and consumption is shown when combining equation (3.7) and equation (3.6). The

difference between this paper and the classical New Keynesian model is I have one more

equation to represent housing demand, which can be found in equation (3.9). In the housing

demand equation, we can see that the marginal utility gain of increasing in housing services is

equal to the marginal utility loss of decreasing in consumption. There are two parts consisting

of marginal utility gain of increasing housing housing services. One is housing services in

the current period. The other is the expected value of housing.

30 Benchmark Model

Firms

On the supply side, as mentioned earlier, there are two sectors: general sector as well as

the housing sector. The general sector and housing sector produce consumption goods and

new houses using capital (Kc,t , Kh,t) and labour (Nc,t , Nh,t). Sticky prices is introduced

in the general sector by assuming monopolistic competition through Calvo-style contracts

and flexible housing price is allowed in the housing sector, which two sectors use different

technologies (Ac,t , Ah,t). As mentioned in Section 3.2.1, there are two reasons why housing

might have flexible prices. First, housing is relatively expensive, which have a bargaining

space on the price of housing. Second, housing production is very sensitive to a monetary

contraction. In the following, I first display some common features in both housing sector

and general sector and then discuss the behaviour of each sector respectively.

The Representative Firm

The general sector and housing sector both hire labour (Nc,t , Nh,t) and buy capital (Kc,t , Kh,t)

to produce consumption goods (Yc,t) and new houses (Yh,t). The technology in different

sectors available to economy is described by a constant-return to scale production function 1:

Yi,t = Ai,tKαi,t−1N1−α

i,t i = c,h (3.10)

where 0 ≤ α ≤ 1 is output elasticities of capital. It measures the responsiveness of output

to the change of capital. Yi,t is consumption goods when i = c and is housing goods when

1I do not include the factor land on the supply side of the housing sector in my research. The reason is Ifocus on the cyclical fluctuations in the Chinese housing market and abstract from the long-run housing pricedynamics that may be related to long-run income and population growth. Land expansion is a proportion of thepopulation growth. According to Deng et al. (2008), they use the empirical study to show that the populationgrowth in China is one of the key variables in the urban land expansion. And also, Deng et al. (2009) rejectthe role that population growth is an important determinant factor in explaining the fluctuation of housingprice in China. In addition, the housing price consists of land price and house value. The land price rise inproportion with population, which is not concerned in my research. I focus on the later one housing value thatis the fluctuation of the housing price.

3.2 Model 31

i = h. Ki,t−1 and Ni,t represent capital and labour in the different sectors. Ac,t measures

productivity in the non-housing sector and Ah,t captures the technology in the housing sector.

As mentioned earlier, the productivity shock in both sectors are assumed to be non-stationary,

which follow a stochastic trend. Therefore, the stochastic process of productivity shock can

be written as:

∆lnAc,t = ρc,t∆lnAc,t−1 + vc,t (3.11)

∆lnAh,t = ρh,t∆lnAh,t−1 + vh,t (3.12)

This specification implies that shocks, vi,t , will have permanent effects on the level of Ai,t .

The firm invest capital following the linear capital accumulation identity.

Ki,t = Ii,t +(1−δk)Ki,t−1 i = c,h (3.13)

where δk is the depreciation rate and Ii,t is the gross investment in the different sector.

Housing Sector

Firms in the housing market operate the perfectly competitive product, which hire labours

and buy capitals to produce new houses. Empirical studies show that the capital stock does

not change very much from period to period. Economists usually rationalise this by assuming

that there are some forms of "adjustment costs" that prevent firms from changing their capital

stock too quickly. Hence, the "capital adjustment costs" is introduced in the firm side so that

to avoid the investment excessively volatility. Assuming there is a convex adjustment cost to

capital facing by the representative firm. I use the quadratic form for tractability.

Φ(.) =κ

2(Kh,t+1 −Kh,t)

2 (3.14)

32 Benchmark Model

The function Φ(.) represents capital adjustment costs, which is assumed to satisfy Φ(0) =

Φ′(0) = 0 and Φ′′(0) > 0. κ captures a multiplicative constant, which affects adjustment

costs. The firms discount future profit flows by stochastic discount factor. The stochastic

discount factor was defined as:

Mt = βt E0u′(Ct)

u′(C0)(3.15)

The reason why the stochastic discount factor written like this is because this is how

households value future dividends. An additional units of utility u′(Ct) is generated at time t

because of one unit of dividend returned to the household, which using β to discount back to

the present period 0. Therefore, the firm maximise the present discounted value of profit,

Vh = E0

∞

∑t=0

Mt [Yh,t ph,t − Ih,t − (wt + εnht )Nh,t −

κ

2(∆Kh,t)

2] (3.16)

subject to the constraints law of motion of the capital stock (3.13) and production function

(3.10) by choosing capital Kh,t and labour Nh,t

Imposing the constraints in each period, the firm’s problem can be re-written as:

maxKh,t ,Nh,t

Vh = E0

∞

∑t=0

Mt [(Ah,tKαh,t−1N1−α

h,t )ph,t − (wt + εnht )Nh,t

−(1+ εkht )Kh,t +(1−δk)Kh,t−1 −

κ

2(∆Kh,t)

2]

(3.17)

where the terms εnht and εkh

t are the labour demand shock and capital demand shock in the

housing sector, which capture other imposts or regulation on firms’ use of capital and labour

respectively. Over the last two decades, China has maintained a rapid economic growth

rate and experienced housing institution reforms. These have significantly affected capital

demand of housing industries and are plausible sources of capital demand shock. The firms

in the housing sector optimally choose capital and labour to maximise their profits. The

3.2 Model 33

demand for labour and capital are represented below:

(1−α)Yh,t

Nh,tph,t = (wt + ε

nht ) (3.18)

Equation (3.18) shows the labour demand of firms in the housing sector, which sets the

marginal product of labour equal to labour price- the real unit cost of labour to the firm wt

and the stochastic shock term εnht .

(1+ rt)[1+κ(Kh,t −Kh,t−1)+ εkht ] =

αYh,t

Kh,tph,t +(1−δk)+κ(Kh,t+1 −Kh,t) (3.19)

Equation (3.19) represents the capital demand of firms in the housing sector. εkht is the

stochastic shock to capital demand. From the above equation, we can see that firms can either

invest 1+κ(Kh,t −Kh,t−1)+ εkht amounts of bonds in period t, which yields a gross return of

(1+ rt)[1+κ(Kh,t −Kh,t−1)] in period t +1 or to get the additional unit of capital (marginal

product of capital) yields AtFK(Kt ,Nt) units of output next periods. Also, an extra unit of

capital reduces tomorrow’s adjustment costs by κ(Kh,t+1 −Kh,t)

General Sector

Production in the general sector is split into two stages, where the final goods stage operate

perfect competition and the intermediate goods stage is monopolistic competition. For the

final goods stage, the general final goods are produced by applying a constant elasticity (CES)

bundler of intermediate goods. The downward sloping demand curve for intermediate goods

producers is obtained through the profit maximisation in the final goods sector operating

competitively. For the intermediates goods stage, the intermediate goods are produced using

the Cobb-Douglas production function. The large number of intermediates producers behave

as monopolistically competitive and have pricing power. The difference between the general

34 Benchmark Model

sector and the housing sector is the intermediate producers in the general sector optimises

along three dimensions, not only capital and labour but also price of intermediate goods. The

intermediate goods firms in the general sector can exploit their market power.

The Final Goods

There are one final goods firm and a continuum of intermediate goods firms (of unit indexed

by k ∈ [0,1]) . The final goods firms behave as perfectly competitive and produce the final

goods at the time t,Yc,t , which aggregates the continuum of intermediate goods in period t,

Yc,t(k) according to the CES production function.

Yc,t =

[∫ 1

0Yc,t(k)

ψ−1ψ dk

] ψ

ψ−1

(3.20)

where there is an assumption: ψ > 1; ψ is the elasticity of substitution among the different

intermediate goods. The integral is raised to the power ψ/(ψ −1) to make the production

function display constant returns to scale.

Final good firms face the problem of profit maximising.

maxYc,t(k)

Pc,tYc,t −∫ 1

0Pc,t(k)Yc,t(k)dk (3.21)

substitute out Yc,t using equation (3.20). The profits will end up with zero since the firm

behaves as perfectly competitive, which total revenue that the final goods price times the

amount of final goods minus total cost that the price of all intermediate goods times quantity.

maxYc,t(k)

Pc,t

[∫ 1

0Yc,t(k)

ψ−1ψ dk

] ψ

ψ−1

−∫ 1

0Pc,t(k)Yc,t(k)dk (3.22)

The first order conditions with respect to Yc,t(k):

3.2 Model 35

Pc,t

[∫ 1

0Yc,t(k)

ψ−1ψ dk

] 1ψ−1

Yc,t(k)− 1

ψ = Pc,t(k) (3.23)

this results in the demand function for intermediate goods k

Yc,t(k) = Yc,t

(Pc,t(k)

Pc,t

)−ψ

(3.24)

this demand function represents that the demand for intermediate goods depends negatively

on its relative price and positively on total production. Substitute out Yc,t(k) using equation

(3.24) into (3.20) comes:

Yc,t =

∫ 1

0

[Yc,t

(Pc,t(k)

Pc,t

)−ψ]ψ−1

ψ

dk

ψ

ψ−1

= Yc,t

[∫ 1

0

(Pc,t(k)

Pc,t

)1−ψ] ψ

ψ−1

(3.25)

rewrite equation (3.25) gives,

1Pc,t

=

[∫ 1

0

(1

Pc,t(k)

)ψ−1

dk

] 1ψ−1

(3.26)

and this results the aggregate price level,

Pc,t =

[∫ 1

0Pc,t(k)1−ψdk

] 11−ψ

(3.27)

The Intermediate Goods Firms

The intermediate goods firms behave as monopolistically competitive, and the Cobb-Douglas

production function is used to produce intermediate goods. They optimise along three

dimensions, not only capital and labour like in the housing sector but also price. The

intermediate goods firms set price following a Calvo rule (Calvo (1983)). That is in each

period, a fraction 1−ω of firms are randomly selected to reset their price for period t, P⋆t (k).

36 Benchmark Model

The rest fraction ω of firms are not able to choose their prices optimally. They keep their

price as same as the last updating.

The intermediate goods firms in the general sector share the similar optimal behaviour

of choosing capital and labour like in the housing sector. The optimal choice of labour and

capital in the general sector are presented below 2:

(1−α)Yc,t

Nc,t= (wt + ε

nct ) (3.28)

Equation (3.28) shows the labour demand of firms in the general sector. The marginal

product of labour equal to its price wt , which is the real wage that is common to all firms in

both sectors. εnct is the labour demand shock in the general sector.

(1+ rt)[1+κ(Kc,t −Kc,t−1)+ εkct ] =

αYc,t

Kc,t+(1−δk)+κ(Kc,t+1 −Kc,t) (3.29)

Equation (3.29) represents the capital demand in the general sector. It shares the

same interpretation in the housing sector. The left-hand side of equation (3.29) gives

the intermediate firms behaviour of investing bonds in period t with the gross return of

(1+ rt)[1+κ(Kc,t −Kc,t−1)] in period t +1. The right-hand side shows the return of getting

the additional unit of physical capital. Therefore, it is equivalent to invest bond or capital.

εkct is capital demand shock in the general sector.

The labour demand and capital demand in the general sector are obtained by maximising

the discounted present value of profits. However, the choice of optimal price is not part of

today’s maximisation problem. The reason is the optimal price that chosen in period t +n is

independent of the price chosen today, which depends on the realisation of the economy from

period t to period t +n and information available in period t +n. There are often two steps to

2The detailed derivation can be found in Housing sector

3.2 Model 37

obtain the optimal price. First, minimise the costs to get marginal cost and then maximise

the market value of intermediate goods firms subject to the demand for their output by the

final goods firm following a Calvo contract (Calvo (1983)).

The firms in the general sector face the cost minimisation problem.

minKc,t(k),Nc,t(k)

[rtKc,t(k)+wtNc,t(k)] (3.30)

subject to the production function, equation (3.10). obtain the marginal cost

mct =1

αα(1−α)1−αA−1

t rαt w1−α

t (3.31)

Price rigidity is introduced in the general sector and follow Calvo (1983) contract. That

implies the firms cannot change their prices freely each period. In particular, in each period

a fraction ω of firms are not able to change its price and has to stick to the price chosen

in the previous period. The rest firms (1−ω) can adjust their prices at time t. A firm is

given the ability to change its price at time t. It adjusts their prices to maximise the expected

discounted value of profits, since it will, in expectation, be stuck with this price for more

than just the current period. In this case, the firm discount factor contains two parts, not only

have the usual stochastic discount factor but also include the probability that firm cannot

change their price. Hence, the firms will discount profits s periods into the future by:

βs u′(Ct+s)

u′(Ct)ω

s (3.32)

where β s u′(Ct+s)u′(Ct)

is the usual stochastic discount factor and ωs is the probability that firm

will be stuck with a price for s periods. If ω is small, then the firms get to update their prices

frequently and thus will heavily discount future profit flows when making current pricing

decisions. On the other hand, if ω is large, it is very likely that a firm will be "stuck" with

38 Benchmark Model

whatever price it chooses today for a long time and thus be relatively more concerned about

the future when making its current pricing decisions.

The intermediate goods firms choose the optimal price in case of the possibility of being

stuck with a price. The firms try to maximise the profit :

maxPc,t(k)

Et

∞

∑s=0

(ωβ )s u′(Ct+s)

u′(Ct)

(Pc,t(k)Pc,t+s

Yc,t+s(k)−mct+sYc,t+s(k))

(3.33)

substitute out Yc,t+s(k) using equation (3.24) gives:

maxPc,t(k)

Et

∞

∑s=0

(ωβ )s u′(Ct+s)

u′(Ct)

(Pc,t(k)Pc,t+s

(Pc,t(k)Pc,t+s

)−ψ

Yc,t+s −mct+s

(Pc,t(k)Pc,t+s

)−ψ

Yc,t+s

)(3.34)

Equation (3.34) shows the problem that intermediate goods firm faced to maximise the

real profits discounted by the stochastic discount factor and the probability of being able to

change the price. The optimal behaviour of choosing price is:

Et

∞

∑s=0

(ωβ )sΛt,t+s((1−ψ)Pc,t(k)−ψP−(1−ψ)

c,t+s Yt+s +ψPc,t(k)−ψ−1mct+sP−(1−ψ)c,t+s Yc,t+s) = 0

(3.35)

where Λ = u′(Ct+s)u′(Ct)

, to make it simply:

Et

∞

∑s=0

(ωβ )sΛt,t+s((ψ −1)Pc,t(k)−ψP−(1−ψ)

c,t+s Yc,t+s)

= Et

∞

∑s=0

(ωβ )sΛt,t+s(ψPc,t(k)−ψ−1mct+sP

−(1−ψ)c,t+s Yc,t+s)

(3.36)

Here come the optimal price P⋆t that replaced Pc,t(k) set by intermediate goods firms:

3.2 Model 39

P⋆t =

ψ

ψ −1Et ∑

∞s=0(ωβ )sΛt,t+s(mct+sP


Et ∑∞s=0(ωβ )sΛt,t+s(P


(3.37)

Each firm updating their price will follow the same optimal pricing behaviour, which

is shown in equation (3.37) since each firm face the same marginal cost and take aggregate

variables as given. This optimal price setting equation will be used to derive the equation for

price dynamics of the model.

As mentioned, there are 1−ω of firms optimal reset their price for period t, P⋆t (k) and ω

fraction of the firms cannot adjust their price following the assumption that keeps their price

as same as in the previous period. Recall the pricing rule of final goods firms (3.27) and split

it into two part: the optimal pricing part and the previous price part. This gives the updating

price level expression:

Pc,t = (1−ω)P⋆t +ωPc,t−1 (3.38)

log-linearisation of optimal price equation (3.37) and updating price level (3.38), Combining

them gives the standard forward-looking New Keynesian Phillips curve:

πc,t = βEt πc,t+1 +(1−ω)(1−ωβ )

ωmct (3.39)

Monetary Policy

Monetary policy is determined by a version of a Taylor rule, which is the central bank reacts

to the inflation. In my research, the central bank reacts to the inflation of consumption goods

and total GDP.

it = i+θπ(πc,t −π⋆)+θGDP( ˜GDPt − ˜GDP⋆

t )+ εmt (3.40)

where following Iacoviello and Neri (2010), GDP consists of the value of consumption goods

and the value of housing. That is GDPt = Yc,t + ph,tYh,t . where ph,t is the steady state real

40 Benchmark Model

housing prices. In consonance with the definition, GDP growth will not be affected when

changing the housing price in the short run. In equation (3.59), it is the target short-term

nominal interest rate. πc,t is the rate of inflation as measured by the GDP deflator, π⋆ is the

desired rate of inflation, i is the assumed equilibrium real interest rate, ˜GDPt is the logarithm

of real GDP, ˜GDP⋆t is a function of productivity in both general and housing sectors.

Market Equilibrium and Identities

The equilibrium of this economy are shown in this section, which solves the different agents’

maximisation problems given the exogenous stochastic processes and initial state variables.

The housing market clears:

Yh,t = Ht − (1−δh)Ht−1 (3.41)

The general market clears:

Yc,t =Ct +(Ic,t + Ih,t)+Gt (3.42)

The Defination of GDP

GDPt = Yc,t + ph,tYh,t (3.43)

Total Labour Demand:

Nt = Nc,t +Nh,t (3.44)

Total Capital Demand:

Kt = Kc,t +Kh,t (3.45)

Fisher Identity:

rt = it −Etπc,t+1 (3.46)

3.2 Model 41

3.2.3 The Linearised Housing Model

The log-linearised housing model is presented in this section. It is convenient for the empirical

analysis when the models described above are linearised. Log-linearisation is a technique

that can convert a non-linear equation into a linear equation, which non-linear model cannot

be solved in a closed form. Moreover, it is quite useful to study economic data using the

logarithm term. I take nature logarithms for the variables in the model. The˜above represents

the logarithmic variables.

Household

The consumption equation is given by:

ct = Et ct+1 −1σc

rt +1σc

(ε pt −Et ε

pt+1) (3.47)

This equation is a traditional forward-looking consumption equation. From this equation,

it can be seen that consumption depends positively on expected future consumption and

negatively on real interest rate.

The housing demand equation is given by:

ht =1−AAσh

Et ph,t+1 −1

Aσhph,t +

σc

Aσhct −

(1−A)σc

AσhEt ct+1 −

1−AAσh

(ε pt −Et ε

pt+1)+

1σh

εht

(3.48)

where A = 1−β (1−δh). From the housing demand equation, it is known that the housing

demand depends on the current and the future relative price of housing and the current and

expected future consumption. It is obviously seen from the equation that no matter the

current and future relative price or consumption, all of them depend on the intertemporal

elasticity of substitution of the housing.

The total labour supply is given by:

wt = η nt +σcct + εlt (3.49)

42 Benchmark Model

Equation (3.49) is labour supply function. The total labour supply is given by the household.

From this equation, we can see that the real wage depends positively on the labour supply

and real consumption.

Housing Sector

The labour demand equation is shown as:

nh,t = yh,t − wt + ph,t + εnht (3.50)

Equation (3.50) is labour demand equation in the housing sector. From this equation, it can

be seen that labour demand in housing sector have a positive relationship with housing price

and housing supply. On the contrary, it depends negatively on the real wage.

The capital demand equation is shown as:

kh,t = k11kh,t−1 + k12Et kh,t+1 + k13yh,t − k14rt + k15 ph,t + εkht (3.51)

Equation (3.51) is the capital demand equation in the housing sector. The coefficients in

front of each variable are calibrated value followed by Meenagh et al. (2010). As we can see

that the expected and past capital, housing output, housing price as well as real interest rate

all affect the capital demand in the housing sector. There is negative effect on real interest

rate and positive effect on other variables.

The housing supply is shown as:

yh,t = α kh,t +(1−α)nh,t + εpht (3.52)

Equation (3.52) is housing supply function. The housing goods are supplied according to a

constant returns to scale production function. The firms use capital and labour to produces

housing goods.

3.2 Model 43

The housing market clearing condition can be written as:

yh,t =HYh

ht − (1−δh)HYh

ht−1 (3.53)

Equation (3.53) is the housing market clearing condition. The total housing supply is equal

to the total housing demand, where HYh

is the proportion of housing demand in the total supply

of housing.

General Sector

The labour demand equation is shown as:

nc,t = yc,t − wt + εnct (3.54)

Equation (3.54) is labour demand equation in the general sector. Similar to the labour demand

in the housing sector, the labour demand in the general sector have positive effect on general

output and negative effect on real wage.

The capital demand equation is shown as:

kc,t = k11kc,t−1 + k12Et kc,t+1 + k13yc,t − k14rt + εkct (3.55)

Equation (3.55) is capital demand equation in the general sector. Similar to the capital

demand in the housing sector, the capital demand in the general sector depends negatively

on the real interest rate and positively on other variables. I follow Meenagh et al. (2010) to

calibrate these parameters.

The price setting equation is given by:

πc,t = βEt πc,t+1 +(1−ω)(1−ωβ )

ωmct (3.56)

44 Benchmark Model

The corresponding marginal cost is given by:

mct = (1−α)wt +α rt − εpnt (3.57)

Equation (3.56) is the standard purely forward-looking New Keynesian Phillips curve. The

above two equations (3.56) and (3.57) show that the current inflation depends on expected

future inflation and the current cost, which the marginal cost is a function of real interest rate,

the real wage and the productivity.

The general goods market clearing condition can be written as:

yc,t = c0ct + k0kt − k0(1−δk)kt−1 + εgt (3.58)

Equation (3.58) is the general goods market equilibrium condition, where c0 is the steady-

state consumption-output ratio, k0 is the steady state capital-output ratio. There are two

kinds of goods produced in the general goods sector: general consumption goods and capital

goods. Therefore, the supply of capital goods can be found in this equation. The general

consumption goods can be consumed by the household and the capital goods can be invested

by firms themselves in both sectors.

Central Bank

The Taylor rule can be expressed as:

it = i+θπ(πc,t −π⋆)+θGDP( ˜GDPt − ˜GDP⋆

t )+ εmt (3.59)

The simply Taylor’s rule is set to achieve both its short-run goal for stabilising the economy

and its long-run goal for inflation. Coefficients θπ and θGDP are assumed to be positively and

chosen by the monetary authority. εmt can be interpreted as monetary policy shock, which

deviates from the steady state due to a change in the policy. A positive εmt can be interpreted

3.3 The Method of Indirect Inference 45

as a contractionary monetary policy shock. On the contrary, a negative εmt represents an

expansionary monetary policy shock.

It should be noticed that the model is loglinearised around a steady state growth path

driven by the growth of the shocks, including the drift term in non stationary productivity.

On the other hand, the level of potential GDP varies with the stochastic trend in produc-

tivity. Therefore, the model is loglinearising around potential GDP which is following the

deterministic trend and stochastic trend.

Equations (3.47) to (3.59) and some identities equations ((3.43) to (3.46)) determine

seventeen endogenous variables in the model. There are eleven exogenous shock variables

in the model driving the stochastic behaviour of the system of linear rational expectations

equations.

3.3 The Method of Indirect Inference

3.3.1 Introduction of Indirect Inference

There are two contributions in this chapter. First, the dynamic stochastic general equilibrium

model is set up incorporating housing sector and some important features of the Chinese

economy. This provides a framework to describe the Chinese housing market in a reasonable

detail. Second, the evaluation and estimation strategy followed Indirect Inference method

using unfiltered non-stationary data are employed in this chapter. The benchmark model

has already discussed in the last section. Therefore, in this section, I focus on introducing

the Indirect Inference method. To be specific, Indirect Inference evaluation and estimation

are going to be introduced in the following. I will emphasise the feature of the unfiltered

non-stationary data in the next section.

Indirect Inference method evaluates the model’s capacity in fitting the data, which

introduced by Minford et al. (2009) and Le et al. (2011) refine this method using Monte

46 Benchmark Model

Carlo experiments. Bayesian also can evaluate the model by creating a likelihood ratio, but

the problem is it can only compare the model with the benchmark model and cannot evaluate

the model against the real data. However, Indirect Inference can provide a classical statistical

inferential framework for testing, which provides a statistical criterion and tells how close

the model is to the actual data. In addition, Le et al. (2016) use Monte Carlo experiments

to compare the power of the Indirect Inference test with the power of the likelihood ratio

test 3. The results show that the power of the Likelihood Ratio test is much lower than

the Indirect Inference test especially in the small sample. The idea of Indirect Inference

evaluation is that the auxiliary model completely independent of the theoretical model is used

to compare the performance estimated on the real data and simulated data. The auxiliary

model in my research is VARX since the unfiltered data are used in evaluation and estimation.

According to Meenagh et al. (2012b), a Vector Error Correction (VECM) model or Vector

Auto Regression with the Exogenous variable model (VARX) can be used as an auxiliary

model if the shocks are non-stationary. Wald test is employed as the criterion when evaluating

the model, which compare the Wald statistic calculated using simulated data with the Wald

statistic calculated using actual data. If the model can pass the test, that implies the simulated

data generated by the model are similar to those generated using the actual data. It shows

that the model can explain the economy properly. For this reason, the behaviour of simulated

data is not significantly different from the behaviour of actual data. If the model fails the test,

Indirect Inference estimation is used to search for a set of coefficients that could improve the

performance of the model.

In terms of indirect inference estimation, which has been widely used in the estima-

tion of structural models (eg.Smith (1993), Gregory and Smith (1991) Gourieroux et al.

(1993)(Gregory and Smith, 1991); Gourieroux and Monfort (1996) and Canova (2007))4.

Indirect Inference estimation is based on the Indirect Inference testing, which repeats the3They use both stationary and non-stationary data to do the comparison.4Recent literature using this method include Minford and Ou (2010), Liu and Minford (2014) and Le et al.

(2014)


testing procedure to find out the global minima of the Wald statistic. The basic idea of

Indirect Inference is trying to search for a set of coefficients that are best able to satisfy the

test criterion. Simulated Annealing algorithm is employed to execute the idea of finding the

minimum Wald statistics.

In summary, the reason why use Indirect Inference evaluation and estimation in my

research is one can test the model unconditionally against the data and can find a certain

set of structural parameters to ensure it to fit the data as closely as possible. On the other

hand, according to Le et al. (2015) the low sample bias feature is another advantage of the

Indirect Inference for small samples. Consider the Chinese housing market does not last for

a long time, the data is limited. Therefore, it is valid and efficient for using indirect inference

method.

3.3.2 Indirect Inference Testing

Indirect Inference is a simulation-based method for estimating the parameters of economic

models. Therefore, it is better to know the testing principle and procedure before estimation.

Testing uses the auxiliary model to compare the actual data with the simulated data generated

from the model. Vector Auto Regression with the Exogenous variable model (VARX) is

employed as an auxiliary model following Meenagh et al. (2012b). The detail of the auxiliary

model will be discussed in the following section. The Wald statistic is used as the criterion

when testing the model, which is the differences between the coefficients from simulated and

actual data. The VAR coefficients β α can get from the actual data and the N sets of VAR

coefficients β i(i = 1 : N) can be obtained from the simulated data, from which we perform

the relevant calculation. The Wald statistic is calculated as following:

W = (β α − β )′Ω−1(β α − β ) (3.60)

48 Benchmark Model

where β = E(β i) = 1N ∑

Ni=1 β i and Ω = cov(β i − β ) = 1

N ∑Ni=1(β

i − β )(β i − β )′

It is interesting to find whether the Wald statistic calculated using actual data lie in the

range of Wald statistic get from the N sets of simulated data. If in that range the model can

pass the test, which means the macroeconomic model is the data generating mechanism.

There are three steps to perform the testing procedure, which originally proposed in

Minford et al. (2009) and Le et al. (2011) refined using Monte Carlo experiments, and also Le

et al. (2016) apply this testing procedure using non-stationary data. A brief testing procedure

is presented in the following.

Step1: Calculate shock processes

The residual and innovation of economic model condition on the data and parameters

are calculated first. If the model equation has no future expectations, the structural errors

can be simply back out from the observed data and parameters of the model. If there are

expectations in the model equations, the rational expectation terms can be calculated using

the robust instrumental variables methods of McCallum (1976) and Wickens (1982). The

lagged endogenous data are used as instruments and hence the auxiliary VAR model are used

as the instrumental variables regression. The errors are treated as autoregressive processes

and use OLS to estimate the autoregressive coefficients and innovations.

Step 2: Simulated data by bootstrapping

Step two is simulating the data. The innovation can be obtained from step one, and

the simulated data can be obtained by bootstrapping these innovations. Bootstrap by time

vector is used to preserve any simultaneity between them and solve the resulting model using

Dynare. This process needs to be repeated so that I can get the N bootstrapped simulations 5,

drawing each sample independently.

Step 3: Compute the Wald statistic5I bootstrap 1000 times in my research.


I use both actual data and the N samples of simulated data obtained from the last step

to estimate the auxiliary model, which can obtain the coefficients of the auxiliary model

from both actual and simulated data. Then, I use Equation (3.60) to calculate the Wald

statistic. According to Le et al. (2011), there are two different types of Wald statistic: the

’Full Wald’ and the ’Directed Wald’. In the Full Wald, all the endogenous variables from

the DSGE model need to be considered in the auxiliary model. It should be noticed that the

more variables and lags are included in the auxiliary model, the higher the chance that the

model will be rejected. Adding more endogenous variables to the auxiliary model will raise

the power of the test, which provides a more stringent test. Therefore, the Directed Wald

statistic is used to focus on some aspects of the model’s performance, which consider some

key endogenous variables in the auxiliary model.

In order to make the model to fit the data at the 95% confidence level, Wald statistic

for the actual data should be less than the 95th percentile of the Wald statistics from the

simulated data. In order to make it easier to understand whether the model has been rejected

by the data, the transformed Wald is introduced. The criteria of rejection is that when the

Wald statistic was equal to the 95th percentile from the simulated data, which the transformed

Wald is 1.645 using Formula 3.61. Then compare the transformed Wald of the actual data

with the criteria (Transformed Wald of 1.645). If the transformed Wald of the actual data is

greater than 1.645, then the model reject by the actual data. If it is less than the criteria, that

means the structure model can replicate the behaviour of the data.

T = 1.648(

√2wα −

√2k−1√

2w0.95 −√

2k−1) (3.61)

where wα is the Wald statistic on the actual data and w0.95 is the Wald statistic for the

95th percentile of the simulated data.

50 Benchmark Model

The above steps show how to test a given model with particular parameter values. These

steps can be also shown graphically. I follow Minford and Ou (2010) to illustrate testing

procedure using the diagram (Figure 3.2) below. Panel A of Figure 3.2 summarises the main

features of Indirect Inference testing I described above. The mountain-shape diagram in

Panel B, replicated from Meenagh et al. (2009), shows that how ‘reality’ is compared to the

model’s predictions using the Wald test when two parameters are considered. In panel B, the

mountain represents the corresponding joint distribution generated from model simulations

and the real data estimation can be either spot. If point A is the real data based estimates, the

theoretical model falls the test because what the model predicts is too far away from what

reality suggests. In contrast, if the real data based estimates ‘on the mountain’,that means the

reality is captured by the joint distribution of the chosen features implied by the model. The

Wald statistic I introduce above formally evaluates these distances.


Fig. 3.2 The Principle of testing using indirect inferenceSource: Minford and Ou (2010)

The above have already shown how a given model with particular parameter values is

tested specifically and expressed the principle using the graph. These parameter values can

get from calibration. However, if the calibrated value is inaccurate, the model would be

probability rejected since the power of the test is high. Therefore, it is necessary to search

for a set of coefficients that can explain the behaviour of data. This is where I introduce

Indirect Inference Estimation. The idea of this estimation is that searching for the numerical

parameter values to minimise the Wald statistic and test the model on these values. The

52 Benchmark Model

model itself is rejected if it is rejected on these values. The detail of Indirect Inference

estimation is going to be introduced in the next section.

3.3.3 Indirect Inference Estimation

The evaluation method using Indirect Inference is to check whether the chosen parameter set

could have generated the actual data. As discussed in the above section, the model would be

probability rejected if the calibrated value is inaccurate. Another set of parameters could pass.

If no set of parameters can be found to pass the test, then the model itself is rejected. Maybe

the model has already unrejected since it has already gotten closer to the data with alternative

parameters. Indirect estimation is used to find the parameters that can minimise the overall

Wald statistic and maximise the chances of the model will not be rejected. The process of

Indirect Inference estimation is simply shown as the following: First, the coefficients are

taken as an input to minimise the object function, then do the testing procedure as mentioned

above. At last, the output is the Wald statistic.

Following Le and Meenagh (2013), a simulated annealing algorithm is chosen as the

minimising algorithm to perform the Indirect Inference estimation, which is a way to imply

the Indirect Inference into practice. Simulated annealing is the physical process of heating

to minimising the system energy by lowering the temperature to decrease defects. It is the

same logic to search for a minimum in a more general system when applying to Indirect

Inference. The algorithm is used when finding the minimum Wald statistics implied by the

real and simulated data. A new state is randomly generated at each iteration of the simulated

annealing algorithm and then decides whether to moving the system to a new state or just

staying in the current state. The distance between new state and the current state is based on

a probability distribution with a scale proportion to the temperature. This leads the system to

move to the states of lower Wald statistic. At last, this iteration will stop when the objective

function is minimised.


The advantage of simulated annealing compare to other methods is the algorithm avoids

become trapped in the local minima and can find globally for more possible solutions. It

repeats the testing procedure to search for the global minima of the Wald statistic. A smaller

Wald statistic compared with any point preceding it in the previous is found at a new point in

the parameter space. The algorithm chooses this current point as a starter to search for the

minimum proceeds. In the following searching procedure, it is normal for the algorithm to

move to points with larger Wald statistic. At last, after a certain number of best points are

found, the search is once again widened by increasing the acceptance probability. There are

different setting for Simulated Annealing. In my research, the bounds are set to be within

40% of the initial calibrated parameters and the maximum number of iterations is set to be

1000.

3.3.4 The Choice of the Auxiliary Model

As mentioned in Section 3.2.1, the technology shock in both housing and general sector are

non-stationary shock and the data used in the evaluation and estimation are unfiltered data.

According to Le et al. (2016), if the data are non-stationary, in order to do the evaluation, an

auxiliary model with stationary errors need to be created. Therefore, VAR with the exogenous

variable is used as the auxiliary model when data are non-stationary. Meenagh et al. (2009)

also mentioned that the VAR model is an approximation of the reduced form of the DSGE

model.

The structural DSGE model after log-linearisation usually can be written as a function:

A(L)yt = B(L)Etyt+1 +C(L)xt +D(L)et (3.62)

54 Benchmark Model

where yt is a vector of endogenous variables, Etyt+1 is a vector of expected future

endogenous variables, xt is an exogenous variable which is assumed to be driven by

∆xt = α(L)∆xt−1 +d +b(L)zt−1 + c(L)εt (3.63)

The exogenous variables xt including stationary and non-stationary shocks like productiv-

ity shocks. et and εt are both i.i.d and the means are zero. xt is non-stationary, yt is linearly

dependent on xt . Therefore, yt is also non-stationary. L is the lag operator Yt−s = LsYt and

A(L), B(L) etc is a matrix polynomial functions in the lag operator of order h that have roots

of the determinantal polynomial lies outside the complex unit circle.

The general solution of yt can be written as

yt = G(L)yt−1 +H(L)xt + f +M(L)et +N(L)εt (3.64)

where f is a vector of constants and polynomial functions in the lag operator have roots

outside of the unit circle. Since yt and xt are both non-stationary, the solution of the model

has p cointegrated relations given by:

yt = [I −G(1)]−1[H(1)xt + f ] = Πxt +g (3.65)

The matrix Π is a p∗ p matrix, which has rank 0≤ r < p, where r is the number of linearly

independent cointegrating vectors. yt − [Πxt +g] = ηt , where ηt is the error correction term.

yt is a function of deviation from the equilibrium in the short run. In the long run, the solution

to the model is given by:

yt = Πxt +g (3.66)


xt = [1−α(1)]−1[dt + c(1)ξ ] (3.67)

ξt =t−1

∑s=0

εt−s (3.68)

where yt and xt are the long run solution to yt and xt respectively. It can be seen

that the long run solution of xt can be decomposed into two components: a deterministic

trend xdt = [1−α(1)]−1dt and a stochastic trend xs

t = [1−α(1)]−1c(1)ξt . There are two

components in the endogenous variables: this trend and a VARMA in deviations from it.

Meenagh et al. (2012a) formulate this as a cointegrated VECM with a mixed moving average

error term, wt .

∆yt =−[I −G(1)](yt−1 −Πxt−1)+P(L)∆yt−1 +Q(L)∆xt + f +M(L)et +N(L)εt

=−[I −G(1)](yt−1 −Πxt−1)+P(L)∆yt−1 +Q(L)∆xt + f +wt

(3.69)

wt = M(L)et +N(L)εt (3.70)

This suggests that the VECM can be approximated by the VARX:

∆yt =−K(yt−1 −Πxt−1)+R(L)∆yt−1 +S(L)∆xt +g+ζt (3.71)

where ζt is an i.i.d with zero mean, since

xt = xt−1 +[1−α(1)]−1[d + εt ] (3.72)

yt = Πxt +g (3.73)

56 Benchmark Model

The VECM can be also rewritten as:

∆yt = K[(yt−1 − yt−1)−Π(xt−1 − xt−1)]+R(L)∆yt−1 +S(L)∆xt +h+ζt (3.74)

According to Le et al. (2016), either equations (3.71) or (3.74) can be used as the auxiliary

model. The equation (3.71) can be rewritten as following:

yt = [I −K]yt−1 +KΠxt−1 +n+ t +qt (3.75)

where the errors qt now consist of the lagged difference regressors and the deterministic

time trend in xt which affect both endogenous and exogenous variables. The equation (3.74)

is used throughout my research followed Le et al. (2016), which distinguishes between

the effect of the trend component and the temporary deviation of xt from the trend. The

advantage is that it is possible to estimate the parameters of equation (3.74) using classical

OLS methods. It also proved by Meenagh et al. (2012a) that this procedure is extremely

accurate using Monte Carlo experiments.

3.4 Data and Calibration 57

3.4 Data and Calibration

3.4.1 Description of Data

Chinese quarterly data over the period 2000Q1 - 2014Q4 are used to do the evaluation and

estimation. A full data description can be found in the Appendix A. In order to match the

variables in the log-linear model, I first convert all the nominal variables to the real term per

capita. Then, I take nature logarithms of the unfiltered observable. Figure 3.3 shows the real

term per capita time series data.

data.png data.png data.png data.png

Fig. 3.3 Chinese macroeconomic data: 2001Q1 to 2014Q4

It can be seen from the Figure 3.3, the Chinese real housing price has been increasing

dramatically since the end of 2002 and is sustained for several years until the recent global

financial crisis. It should be noted that there is a significant drop after 2007. The housing

price rises immediately and reaches its peak in 2009. It also can be seen that the growth rate

become less immoderate after 2009.

58 Benchmark Model

3.4.2 The Advantage of Non-stationary data

As mentioned earlier, one of the important contribution made in this research is that the

unfiltered data are used in the evaluation and estimation. The business cycle model focus on

studying the dynamics and the choice of macroeconomic policy on stabilising the fluctuations,

which try to abstract from the long run uncertainty economic surrounding. Most researchers

use some technique such as the Band Pass (BP) and the Hodrick-Prescott (HP) filters to

abstract the uncertainty trend. However, there is a criticism of both the HP filter and BP filter.

According to Harvey and Jaeger (1993), the HP filter can lead to spurious cyclical behaviour.

Moreover, Cogley and Nason (1995) and Murray (2003) study the spurious dynamic causing

from HP and BP filters to non-stationary data and show that these filters cannot distinguish

between difference-stationary and trend-stationary. The HP and BP filters are a mathematical

tool used in the business cycle, which are not based on theories. Hence, the precision of

the driving process that leads to trend behaviour cannot be identified using these techniques.

Some researchers maybe consider linear detrend to make the time series stationary. However,

the problem of this method is some data cannot be stationary even if they have already

linear detrended. According to Canova (1998), the linear detrend maybe not accurate when

the data have a stochastic trend since it cannot isolate fluctuations. On the other hand, the

non-stationary data used in the model could shed light on the economy in a significant way

where the stationary data do not. Meenagh et al. (2012b) take the recent Great Recession as

an example to express this idea.6 Le et al. (2014) also find this stylised fact in China.

In summary, the advantage of using non-stationary data would be twofold: On the

one hand, the filters cannot provide an appropriate and precise decomposition into a non-

stationary time series. The business cycle dynamics can be generated using the filters even

if they are not present in the original data. On the other, I am interested in the stochastic

trend, which arises from the unit root processes of the technology shocks. I want to keep

6There is a severe decrease in the OECD after the crisis. However, the trend level of GDP still cannot reachits previous level.


the non-stationarity and do not remove it since the non-stationary data could explain some

dynamic properties of the model that stationary data could not.

3.4.3 Calibration

In this section, the calibrated values in my research are introduced before evaluating and

estimating. The parameter values are partitioned into two groups. The first group of

parameters conduct the dynamics of the model, which can be found in Table 3.1. These

parameter values are calibrated according to the previous literature and the observations in

some empirical analyses. For example, the coefficients of the Taylor rule and the quarterly

depreciation rate of housing and capital. The second group of the parameters are the steady

state of the model, which is shown in Table 3.2. These kinds of parameter values used in this

chapter are obtained from the data. For example, the consumption output ratio or residential

investment ratio. I am going to test the model with these calibration values in the following.

If the model cannot pass the test, the Indirect Inference estimation will carry out to reestimate

these parameters. In the estimation, I fix these parameters (β ,δk,δh) because the accounting

information is used to identify them.

On the households side, β is set to be 0.985, using Chinese quarterly data. This is in line

with the standard in the most DSGE housing literature, which implies a steady state annual

real interest rate of around 4 percent7.

σi denotes the coefficient of relative risk aversion of households. The elasticity of

intertemporal substitution is given by 1/σi, which measures the responsiveness of the growth

rate of consumption to the real interest rate. The range of σi, according to Gandelman and

Hernández-Murillo (2015) , is between 0 and 3. I calibrate coefficient of relative risk aversion

for consumption (σc) at 2 in the general sector according to Walsh (2003) and 1 (σh) in the

housing sector in accordance with Iacoviello (2005), implying the elasticity of intertemporal

7Using steady state R = 1β

to get β

60 Benchmark Model

substitution of 0.5 in the general sector and 1 in the housing sector. In general, a low value of

σi (high intertemporal elasticity) means that consumption growth is very sensitive to changes

in the real interest rate. I calibrate σh is lower than σc since substitution of housing goods is

relatively more sensitive compared to the substitution of general consumption goods when

changing in the real interest rate.

η is the inverse of elasticity of labour supply, that is η = 1/ξ , where ξ is the Frisch

elasticity of labour supply. The Frisch elasticity of labour supply measures the elasticity of

labour supply with respect to wages, which captures the substitution effect of a change in the

wage on labour supply. I follow Iacoviello and Neri (2010) to set the inverse of elasticity of

labour supply at 0.5, which implies the elasticity of labour supply at 2. The lower inverse of

elasticity of labour supply makes the labour supply elastic.

On the firm side, I follow Liu and Ou (2017) to set quarterly depreciation rate of

housing and capital (δh and δk) equaling to 0.015 and 0.03 respectively. It implies an annual

depreciation rate of around 6% in housing and 12% in general capital.

Following Liu and Ou (2017)’s study, the capital-output elasticity α in the Cobb-Douglas

production function is set to 0.3, which is consistent with previous literature. I calibrate price

rigidity ω at 0.84, which in line with Zhang (2009) who employ Chinese quarterly data to

estimate the New Keynesian Phillips curve using GMM. The higher ω implies the longer

durations between price changes. The capital demand coefficients in both sectors getting

from Meenagh et al. (2010) are 0.51, 0.47, 0.02, 025, 0.5 respectively.

In terms of the monetary policy rule, The standard value of θπ=1.5 and θGDP=0.125 are

chosen in line with Taylor (1993). The coefficient of interest rate response to inflation (θπ ) is

set greater than one, which satisfies the Taylor principle. I follow Smets and Wouters (2003)

to set the persistence parameters for different shocks of the exogenous processes, which most

of them are chosen to be 0.85.


Table 3.1 Calibrated Coefficients - Benchmark Model

Defination Parameter Calibration

Households:

Elasticity of substitution of normal goods consumption σc 2

Elasticity of substitution of housing goods consumption σh 1

Inverse of elasticity of labour η 0.5

Household’s discount factor β 0.985

Firms:

Price rigidity ω 0.84

Output elasticity of capital α 0.3

Quarterly depreciation rate of housing δh 0.015

Quarterly depreciation rate of capital δk 0.03

Capital demand coefficients k11, k12 0.51, 0.47


Capital demand coefficients k15 0.5

Monetary Policy:

Taylor Rule response to inflation θπ 1.5

Taylor Rule response to output θGDP 0.125

62 Benchmark Model

Table 3.2 Steady state ratios- Benchmark Model

Defination Parameter Data

Consumption Ratio C/Y 0.38

Investment Ratio I/Y 0.45

Capital Ratio in General sector Kc/K 0.78

Capital Ratio in Housing sector Kh/K 0.22

Labour Ratio in General sector Nc/N 0.79

Labour Ratio in Housing sector Nh/N 0.21

Residential Investment ratio Yh/GDP 0.03

3.5 Empirical Results

3.5.1 Estimation

Model fit

The evaluation and estimation strategy have already been introduced in Section 3.3. Cali-

bration value of the parameters will be used in the testing, choosing VARX as an auxiliary

model as mentioned in Section 3.3.4. The choice of variables in the auxiliary model was

straightforward. The total output is essential to include in the auxiliary model. No matter

what kind of macro model is, it should at least be able to explain the behaviour of output.

One of the main contributions of this thesis is to evaluate whether the DSGE model with

the housing sector would be able to capture the features of the housing market in China.

Hence, the housing variables need to be considered in the auxiliary model. Housing price

is chosen as the second variable in the auxiliary model since it is the most concern housing

variable. The third variable is the interest rate. The reason I choose it in the auxiliary model

is the change in interest rate would affect the housing market that I concerned. Therefore,

3.5 Empirical Results 63

the three variables used in the auxiliary model are: GDP, housing price and interest rate. If

the model does not pass the test, the calibrated value can be predefined as starting value to

implement Indirect Inference estimation. Hence, in this section, I am going to discuss the

Indirect Inference empirical results. Table 3.3 shows the model coefficients of calibration

and estimation together with the testing results that presented in the form of transformed

Wald and P value.

In terms of evaluation, the testing results show in the bottom of Table 3.3, which represents

that the structural model with the calibrated parameters cannot explain the data behaviour.

More specifically, the Transformed Wald statistic for variables GDP, housing price and the

interest rate is 7.41 that greater than the critical value 1.645. Also, the P-value is 0 when

the model employs calibration value, which shows that the model is severely rejected. This

indicates that the structural model does not perform well in generating the observed data

using calibrated parameters. The reason might be either the unreasonable values for some

parameters or the failure of the structural model. Therefore, it is necessary to search for

the numerical parameter values that minimise the Wald statistic and then test the model on

these values. This is why Indirect Inference estimation is employed. The bottom of the last

column in Table 3.3 shows the transformed Wald statistic of the estimated coefficients. The

results show that the estimated model using Indirect Inference method can fit the data well

according to the transformed Wald 1.02 comparing with the critical transformed Wald of

1.645. The p-value of 0.11 also verifies the finding.

In terms of estimated results, the last column in Table 3.3 provides the best fit of coeffi-

cients values. It should be noticed that all coefficients are allowed to change except quarterly

household’s discount factor (β ), the depreciation rate of capital and housing (δk, δh) since

other information are used to identify them.

All of these coefficients have moved within the 40% interval of the initial calibration

value. On the household side, the estimated elasticity of substitution of normal goods

64 Benchmark Model

consumption (σc) has jumped to 2.65. The higher value of σc implies the lower intertemporal

in general sector, which means the consumption is relatively insensitive to the change in the

real interest rate compared with the calibration value. In the housing sector, the elasticity of

substitution of housing goods consumption (σh) decreases to 0.73 , implying that the housing

consumption growth is much more sensitive to changes in the real interest rate than that in

calibration. These estimated elasticity coefficients are in line with the assumption, which the

consumption is very insensitive in the general sector and sensitive in the housing sector to the

change in the real interest rate in China. The estimated coefficient of the inverse elasticity of

labour supply (η) is 0.41 lower than its calibrated value of 0.5, which implies the elasticity

of labour supply is around 2.5. This makes the labour supply more elastic comparing with

the calibration value, which implies that the workers in China are more willing to smooth

working hours when the wage rate change.

On the firm side, the price stickiness ω adjusts to 0.52 after estimation, lower than the

calibration value. That implies the data suggest a lower degree of nominal rigidity in China,

which shows short durations (around two quarters) between price changes. This estimation

result is similar to the finding in Liu and Ou (2017) who provide the estimated price rigidity

ω=0.41. The value of the share of capital in production is significantly higher than what

was initially thought, a value of 0.66 is much closer to Zhang (2009) in their work. For

the capital demand coefficients, the coefficients k11 is lower than the starting value of 0.51,

which implies lower adjustment cost. The higher value of estimated k12 in the housing sector

implies that a lower discount rate of capital. The estimated coefficients in the capital demand

equation satisfy that the sum of k11, k12 and k13 is equal to 1.

For Taylor rule, monetary policy is estimated to be more responsive to inflation and

output fluctuation. More specifically, the inflation response θπ increase from 1.5 to 1.88,

which satisfies the Taylor principle. That implies when interest rate change one unit, the

reaction of inflation will be greater than one-for-one.


Table 3.3 Model Coefficients: 2000Q1-2014Q4

Defination Parameter Calibration Estimation

Households:

Elasticity of substitution of consumption σc 2 2.65

Elasticity of substitution of consumption σh 1 0.73

Inverse of elasticity of labour η 0.5 0.41

Household’s discount factor β 0.985 0.985

Firms:

Price rigidity ω 0.84 0.52

Output elasticity of capital α 0.3 0.66

Quarterly depreciation rate of housing δh 0.015 0.015

Quarterly depreciation rate of capital δk 0.03 0.03

Capital demand coefficients k11, k12 0.51, 0.47 0.30, 0.69


Capital demand coefficients k15 0.5 0.86

Monetary Policy:

Taylor Rule response to inflation θπ 1.5 1.88

Taylor Rule response to output θGDP 0.125 0.25

Trans-Wald (GDP, HP, R) 7.41 1.02

P-value 0.00 0.11

66 Benchmark Model

Error Properties on Non-stationary Data

Unfiltered data are used when doing the evaluation and estimation. The testing and estimation

results show that the structural model that integrates the housing sector can perform well in

generating the observed data. Hence, in this part, the error properties on non-stationary data

are analysed. I follow Le et al. (2014) idea to analyse the error properties. To do this, the

shocks are backed out of the model using unfiltered data and fit to each an AR time-series

process over the period. There are two different types of stationarity test for each calculated

shock process. That is Augmented Dickey-Fuller (ADF) test and the Kwiatkowski Phillips

Schmidt Shin (KPSS) test. The ADF test tests the null hypothesis of the unit root against the

stationarity. On the contrary, the KPSS test evaluates the null hypothesis that the shock is

stationary against the alternative hypothesis that the shock follows a unit root process. Table

3.4 reports the stationarity of each shock and Table 3.5 presents the AR parameters.

It should be noted that the null hypothesis of a unit root cannot be rejected at 5% level

for productivity shock in both housing and general sector under the ADF test. Also, the

KPSS test verifies this finding that all the shocks fail to reject the stationary apart from

the productivity shock in both sectors. That means the productivity shocks in both sectors

contains a stochastic trend, which cannot be the deterministic trend stationary. Hence, the

productivity shocks in both sectors are specified in first differences. This provides evidence

to support modelling productivity shock as the non-stationary shock.

For some shocks like government spending shock, preference shock, labour demand

shock in housing sector and capital demand shock in both sectors cannot be rejected the null

hypothesis of a unit root at 5% significant level using the ADF test, but the KPSS testing

results show that it cannot reject the null of stationarity at 10% significant level. Hence, these

shocks are treated as either stationary or trend stationary due to the theoretical grounds in

line with the setup in Le et al. (2014).


For labour supply shock, housing demand shock, monetary policy shock and labour

demand in general sector no matter the P-value from the ADF test or the statistics from the

KPSS test all imply that these shocks processes are stationary. In the previous literature,

some works using different specification for the exogenous processes. For example, Smets

and Wouters (2007) employ an ARMA mark-up shock to help to fit the model well and also a

higher-order autoregressive process used as the government spending shock in Del Negro and

Schorfheide (2009). In my research, apart from the productivity shocks in different sectors,

the other exogenous shocks processes follow AR(1) dynamics or AR(1) dynamics with a

deterministic trend. The AR coefficients are determined by the estimation process, which

also can be found in Table 3.5. Most of the exogenous process are highly persistence.

Fig. 3.4 Structure Shocks

68 Benchmark Model

Table 3.4 Stationarity of Residual

Shocks ADF p-value KPSS Statistic Conclusion

Government spending 0.4593 0.238410 Trend Stationary

Preference 0.1980 0.195231 Trend Stationary

Labour supply 0.0497** 0.180101 Trend Stationary

TFP – General sector 0.9993 0.955905*** Nonstationary

TFP – Housing sector 0.7184 0.885958*** Nonstationary

Housing Demand 0.0246** 0.120453 Stationary

Monetary policy 0.0327** 0.061916 Stationary

Labour demand - General 0.0451** 0.213222 Trend Stationary

Labour demand - Housing 0.1381 0.236403 Trend Stationary

Capital demand - General 0.1401 0.209563 Trend Stationary

Capital demand - Housing 0.1801 0.179087 Trend Stationary

Notes:1. p-value with ** rejects the unit root process at 5%2. KPSS with ** (***) rejects stationarity at 5% (1%)


Table 3.5 Estimated Shocks Coefficient

Shocks Estimated Coefficient

Government spending 0.9292

Preference 0.8805

Labour supply 0.9569

TFP – General sector -0.6054

TFP – Housing sector 0.6399

Housing Demand 0.5938

Monetary policy 0.6378

Labour demand - General 0.8914

Labour demand - Housing 0.9997

Capital demand - General 0.8692

Capital demand - Housing 0.9301

3.5.2 Properties of the Model

In Section 3.5.1, I have already discussed that the structural model with the estimated

parameters does perform well in generating the observed data. Therefore, in this section, the

estimated model is used to address one of the research questions that raised at the beginning

of this chapter. That is what determines the housing prices in China. In the following, the

variance decomposition of the main variables, shock decomposition of housing price and

impulse response functions of different shocks are employed to analyse the housing price

fluctuation in China.

70 Benchmark Model

Variance Decomposition

Variance decomposition of forecast errors investigates how much of the forecast error variance

of each of the variables can be explained by exogenous shocks to the other variables. Table 3.6

reports the variance decomposition of forecast errors of total output, inflation, consumption,

housing price and the interest rate at different horizons using estimated coefficients.

In terms of housing price, in the short run (1 year), the dominant driving force of housing

price fluctuations is capital demand shock in the housing sector, which explain 65.59% of

the variance in housing price. The housing demand shock also contributes to the volatility of

housing price, which accounts for 18%. By contrast, monetary policy shock only accounts

for 9.90% in the short run. In the long run (10 years), the influence of capital demand shock

and housing demand shock decrease, which the former explains 51.65% while the latter

explains only 10.32%. By contrast, the influence of technology shock in the general sector

on housing price increase, which accounts for 21.26%

As mentioned previously, the capital demand shock in housing sector captures the

regulation on firms’ use of capital. That implies the change in regulation on supply side related

to capital usage affects the housing price dramatically. Chinese housing market experiences

substantial reforms starting in 1998, the full marketisation reform promoted the privatisation

of housing. A series of regulation changes on the supply side are plausible sources of capital

demand shock. Houses were treated as welfare before reform, which the Chinese government

control the housing construction. There is no housing market at that time. However, after

reform, houses were treated as a commodity at prices determined by the market, which can be

purchased or rented. In order to boost housing market development, a series of reforms were

established on the supply side when housing has been commodified, which develop some

new regime of capital accumulation such as expanding production capacity and attracting

foreign investment in economic development. This reform witnessed the Chinese housing

market boom and the Chinese housing industry have become a pillar industry in the Chinese


economy. According to Wu (2015), he believed that capital accumulation is a causative

factor in the growth of the housing market. The change in capital accumulation is linked

with related regulation changes8. These regulations on the supply side played a key role in

housing development.

For the real consumption, the variance decomposition shows that intertemporal preference

shock and technology shock in the general sector play a significant major role. The volatility

of real consumption is mainly driven by intertemporal preference shock. The shock explains

47.74% of the variance in real consumption in the short run (1-year forecast horizons) and

explains 13.49% in the long run (10-year forecast horizons). Technology shock in the

general sector also contributes significantly and explains 22.06% of the variance in real

consumption in the short run. However, in the long run, technology shock in the general

sector becomes more important and explains 68%. The reason why the preference shocks

play an important role in explaining the real consumption in the short run due to the inter-

temporal Euler equation, which directly affects the real consumption. We know from the

model that the productivity technology shock is the non-stationary shock, which implies

it has a permanent effect on output, consumption, capital and investment. Therefore, this

explains why productivity shock contributes significantly to explaining real consumption in

the long run.

The following variables in Table 3.6 GDP, inflation, and interest rate are non-housing

variables. The demand shock (housing preference) in the housing market plays a role in

driving the cyclical fluctuations, which account for over 10% of most variables in the short

run. The supply shock (housing technology), however, plays a minor role in driving the

cyclical fluctuations, which accounts for less than 5% of most variables. However, The

contribution of productivity shock in the general sector to the fluctuations in non-housing

variables is more important especially for the most non-housing variable. Technology shock

explains 37.87% of the variances in GDP in short tun and about 67.66% in the long run.

8Regulation change such as economic decentralisation, fiscal reform etc.

72 Benchmark Model

Tabl

e3.

6V

aria

nce

Dec

ompo

sitio

nof

aggr

egat

eva

riab

les-

Ben

chm

ark

Mod

el

Var

iabl

eG

over

.Sp

end.

Inte

rt.

Pref

.L

ab.

Supp

lyTe

ch.i

nG

ener

alTe

ch.i

nH

ouse

Hou

seD

eman

dM

onet

.Po

licy

Lab

.D.

Gen

eral

Lab

.D.

Hou

seC

apit.

D.

Gen

eral

Cap

it.D

.H

ouse

1Ye

ar:

GD

P0.

070.

841.

1937

.87

1.58

22.1

827

.07

0.04

0.84

1.01

7.31

Infla

tion

15.4

023

.40

13.9

00.

610.

5116

.00

21.1

30.

377.

490.

091.

10

Con

sum

ptio

n0.

7047

.74

1.03

22.0

63.

363.

978.

890.

020.

401.

6310

.20

Hou

sing

Pric

es0.

200.

270.

081.

800.

7018

.80

9.90

0.00

0.13

2.53

65.5

9

Inte

rest

rate

3.82

1.56

0.19

13.6

61.

0355

.19

9.66

0.00

0.04

1.26

13.5

9

5Ye

ar:

GD

P0.

370.

640.

8360

.56

2.05

13.3

516

.14

0.02

0.76

0.63

4.62

Infla

tion

19.3

615

.85

16.2

52.

690.

8310

.75

13.7

80.

2713

.33

0.29

6.61

Con

sum

ptio

n2.

4728

.44

3.80

37.7

23.

715.

088.

040.

050.

250.

949.

50

Hou

sing

Pric

es1.

040.

690.

496.

021.

8014

.08

8.77

0.01

2.78

1.59

62.7

3

Inte

rest

rate

3.25

2.46

1.04

35.3

81.

4435

.58

6.90

0.01

0.68

1.03

12.3

2

10Ye

ar:

GD

P0.

360.

520.

6667

.66

2.63

10.5

312

.73

0.02

0.72

0.50

3.67

Infla

tion

14.1

911

.09

12.9

420

.15

2.22

7.51

9.62

0.19

15.8

90.

295.

92

Con

sum

ptio

n1.

4113

.49

2.61

68.5

62.

312.

443.

840.

030.

170.

454.

70

Hou

sing

Pric

es0.

780.

560.

3821

.26

2.36

10.3

26.

430.

015.

091.

1751

.65

Inte

rest

rate

3.35

2.61

1.56

31.8

21.

3936

.60

7.11

0.02

1.72

1.07

12.7

7


Historical Decompositions

The historical decomposition measures the contribution of each shock to the volatility of

variables over a certain sample period. Figure 3.5 decompose the real housing price over the

shocks during 2000Q1 to 2013Q4. There are some common features when decomposing the

real housing price comparing with those of the forecast error variance decompositions.

In order to make the figure clear, I classify these 11 shocks (government spending shock,

preference shock, housing demand shock, labour supply shock, productivity shock in both

housing and general sectors, capital demand as well as labour demand in both sectors and

monetary policy shock) into 6 categories. The preference shock and labour supply shock

belong to private shocks. The public shocks contain labour demand shock in different sectors

and capital demand shock in the general sector and housing sector. The policy shocks consist

of monetary policy shock and government spending shock. The total factor productivity

(TFP) include the technology shock in two sectors. The housing demand shock does not

classify into private shocks category as I want to observe how this shock influence the real

housing price.

From Figure 3.5, I find that the fluctuation of housing price is mainly driven by the

technology shock, public shocks as well as housing demand shock, which in line with

the finding in the variance decomposition. As can be seen in Figure 3.5, the housing

price experienced a sharp increase at the middle of 2005 and public shock made a major

contribution to the surge in the real housing price. An explanation of this sharp increase

is the abolishment of welfare housing provision and adoption of a more radically market-

oriented approach to housing provision at the beginning of 2005. Housing commodification

accelerates the development of housing industries, that implies more capital are needed.

The injection of capital has to lead to a new housing market cycle, which in line with Wu

(2015). It also supports the finding in variance decomposition that capital demand shock is

the main driving force of the fluctuation of real housing price. A series of housing tightening

74 Benchmark Model

policies were issued to slow down the housing price increase at starting from 2006. These

policies affect the sharp increase in the real housing price but the housing market did not

experience a sharp downturn before the global financial crisis. Hence, households needed to

think carefully about housing purchase. However, the housing boom was hit hardest in late

2007. The outflow of capital caused by the boom of the stock market in 2007. Households

engaged in the stock market. Hence, the demand of housing decrease. In the following, the

housing market also hit by the global economic crisis in 2008. One of the approaches that

the central government attempt to stimulate economic growth after the crisis is using housing

industries as a booster. The stimulation package includes investment in the housing market,

a reduction in interest rates and enhancing capital liquidity. Some new regimes of capital

accumulation such as expanding production capacity and attracting foreign investment were

launched to encourage the housing market. According to Wu (2015), the minimum capital

requirement for commodity housing projects was reduced from 30% to 20%. This is shown

clearly from the graph, in which the rapid rebound following the temporary slowdown. After

2010, the housing price started to fall down. A series of property purchases restrictions and

low productivity, I believe, is the main culprit.

Fig. 3.5 Shock Decomposition - Real Housing Price


Impulse Response Functions

Impulse response functions obtained from the estimated model are shown in this section.

There are three main shocks I concern about: housing demand shock, monetary policy shock

and technology shock in the general sector. In order to study the reaction to these shocks

in different sectors, in the first row I report the response of main model variables in the

general sector and the response of main model variables in the housing sector are shown in

the second row. The third row shows the total GDP and interest rate. The other shocks can

be found in Appendix B

The effects of housing demand shock

Figure 3.6 shows how these key macro variables behave in different sectors when there is

a positive housing demand shock (15.5 standard error). It is assumed that the shocks εht

following the exogenous process lnεht = ρh lnεh

t−1 + vh,t . I focus on the housing sector first.

A positive housing demand shock leads to a significant increase in housing demand, which

lead to the increments in housing price. Supply of housing also rises to meet the high demand

for housing. The housing boom not only affect the housing sector, but the general sector

is also been affected. The housing boom leads to the expansion of output in the general

sector and inflation, which slightly increase compared to those in the housing sector. And a

subsequent increase in interest rate. The impulse responses of selected variables to housing

demand shock are in line with the main findings in many previous literature.

As mentioned in the model part, housing demand shock can be think as the change of

preferences on housing due to social, institutional or income changes. The housing institution

reforms can be explained as the important institutional changes. The private housing market

was established after a series of reforms. A series of reforms were launched, and the welfare-

oriented housing provision was abolished. These lead to the increments in housing demand.

In order to meet the extra demand, the supply of housing raised. Overall, the housing

76 Benchmark Model

Fig. 3.6 Impulse responses to Housing Demand Shock

reform stimulate the whole economic growth, which is a significant fundamental economic

change. On the other hand, China has experienced economic transformation involving fast

productivity progress. According to Bian and Gete (2015), the higher income the household

obtain benefit from higher productivity, the higher demand for housing they desire.

The effects of the monetary policy shock

Figure 3.7 plots impulse responses to the monetary policy shock (0.45 standard error) to

the economy. A positive shock to the monetary policy decrease all the variables in different

sectors. The standard interest rate channel of monetary policy transmission show that such

monetary contraction discourages the investment and consumption so that decrease the total

output. The decrease of consumption shifts the pressure through changes in inflation, which

lower the inflation as well as housing price. That means a tightening of monetary policy

can have a drop in housing price. This maybe due to the Taylor Rule in the model affect

the consumption through Euler equation. This leads to influence the demand of housing,


Fig. 3.7 Impulse responses to Monetary Policy Shock

which triggers the housing price. These results are consistent with the main finding of many

previous literature. A tightening of monetary policy shock decreases all components of real

aggregate demand and real housing prices.

The effects of technology shocks in the general sector

Figure 3.8 plots the impulse responses of model variables to a 3.20 standard error total

factor productivity shock. It should be noticed that the productivity shock in both sectors

follow a unit root process. This non-stationary process has the permanent effect on some

macroeconomic variables such as total output, output in different sectors, real consumption,

housing demand and the stock of physical capital. These variables become more persistent

afterwards lasting over 40 quarters. In response to a positive productivity shock in the

general sector, the supply of output in the general sector goes up due to superior technology.

The increase in supplying lead to the decrease in inflation. The interest rate decrease as a

results of falling inflation. In the housing sector, we can see that the increase of housing

78 Benchmark Model

Fig. 3.8 Impulse responses to Technology shocks in general sector

supply decrease the housing price at the very beginning. As the demand for housing increase

significantly after period 5, the housing price pushes up at the same period.

3.6 Conclusion

The aim of this chapter is focusing on answering the first research question: What is the

driving force in the fluctuations of housing price. The estimated DSGE model with explicit

modelling of the price and quantity of the housing sector and non-stationary productivity

shock is used to study housing market fluctuations in China. Indirect Inference method

is employed to find a right model to explain the data behaviour in the Chinese housing

market. The testing results show that the model using the calibration value is rejected by the

data. Hence, the Indirect Inference method is used to estimate the model over the period

2000-2014, which find out a set of coefficients that can past the test. The model can fit the

data well when a variety of endogenous variables are added to the auxiliary model, explaining

the output, housing price and interest rate that I concerned about. Once find the right model

3.6 Conclusion 79

that can perform well in explaining the data, I discover the housing market using this model.

In terms of the driving force of fluctuations in the Chinese housing market, the variance and

shock decomposition suggest that the capital demand shock plays a significant major role in

explaining the housing price. That maybe because the housing market reform stimulates the

Chinese housing industry, which develops some new regime of capital accumulation. These

regulation change on the supply side played a key role in housing development.

Chapter 4

Model with Collateral Constraint

4.1 Introduction

In the last chapter, I have established a benchmark model with a rich set of shocks to explain

the behaviours and the sources of fluctuations in the Chinese housing market using Indirect

Inference method. The testing results show that the estimated benchmark model can fit

the Chinese data well on the one hand, and that encompasses most of the views on the

sources and propagation mechanism of business cycles on the other hand. The increasing

interest in the DSGE housing model literature have focused on the collateral constraint

on the households’ side, which treats as a channel that connects the housing market to

the wider economy. Previous studies emphasise the role play as the housing collateral in

the households’ optimal decision. They add the channel by splitting the households into

two types: patient (lenders) and impatient (borrowers). The impatient households in the

economy face a binding collateral constraint when participating in loan and mortgage market.

Therefore, the collateral constraint is a channel to amplify the collateral effect on households

borrowing to the whole economy.

One important work is that Iacoviello and Neri (2010) present a Bayesian estimated

DSGE model to study the housing market and business cycle, which emphasise the spillovers

82 Model with Collateral Constraint

effect from the housing market to the wider economy by adding the collateral constraint on

the households’ side. There are different ways to quantify housing market spillovers. The

Iacoviello type model focus on one aspect of spillovers, that is, the relationship between

housing wealth and non-housing consumption. Inspired by Kiyotaki and Moore (1997), these

Iacoviello type model feature a collateral constraint on the households side. That implies

the borrowing capacity of impatient households is limited by a fraction of the expected

present value of total assets such as houses, lands and capitals. In their model, the borrowing

constraint can be thought as a channel to connect the housing market and the rest of the

economy, allowing for ’spillover’ from one sector to the other through that channel.

Iacoviello (2005) constructs a DSGE model including collateral constraints tied to real

estate values for the impatient households. In the extended model, both firms and impatient

households face the credit constraint. The model is used to explain both the business cycle

facts and the interaction between asset prices and economic activity. Then, Iacoviello and

Neri (2010) extend the work of Iacoviello (2005) and present a Bayesian estimated DSGE

model. They show that collateral effects on households borrowing amplify the response

of non-housing consumption to given changes in fundamentals, thus alter the propagation

mechanism.

The transmission mechanism of collateral constraint in Iacoviello type model work as

following. When there is a positive demand shock, the demand for housing rise, housing

price also increases. The rise in asset prices increases the borrowing capacity of the debtors.

That implies they can borrow more due to the high asset prices, allowing them to spend

and invest more. The change in investment will cause the output to fluctuate, which in turn

influences the current asset price. Therefore, a significant transmission channel is generated

through the dynamic interaction between the credit constraint and asset prices.

Based on their analysis framework, more types of shocks and frictions are introduced

to study the housing market. Ng (2015) employ Iacoviello type model to study the sources

4.1 Introduction 83

and consequences of the fluctuations in the Chinese housing market. In terms of the nature

of shocks driving housing price dynamic, they find that housing demand shock explains the

majority of the fluctuations in housing price. In terms of spillover effect work through the

collateral constraint, there is not a unique way to quantify the effect, which depends on the

nature of shocks. Housing demand shock has a larger contribution to the spillover effect

compared to the technology shock. However, the technology shock plays a negligible role in

the spillover effect. Liu and Ou (2017) also use a DSGE model with a collateral constraint

to study the Chinese housing market. Apart from investigating the main driving force of

housing market fluctuation, they also study the housing market spillovers effect in China.

They find that there is a weak spillover effect from the housing market to the wide economy.

He et al. (2017) employ a Bayesian DSGE model with collateral constraints to investigate the

interaction between the housing market and the business cycle. They find that the collateral

constraint plays a significant role in explaining the fluctuate of the business cycle in China,

which amplifies the impact of various economic shocks.

While these studies have highlighted the collateral constraint on households borrowing, to

date there has been no evaluation of the general equilibrium model with collateral constraint.

This is the perspective adopted here. Indirect Inference evaluation is used to check whether

the DSGE housing model with a collateral constraint on the household side can explain the

Chinese housing market well. From the modelling point of view, my starting point is the

benchmark model that introduced in Chapter 3, which a DSGE model include the explicit

modelling of the price and quantity of the housing sector with non-stationary productivity

shock. In order to examine whether the model with collateral constraint can explain the

Chinese housing market well, I include another feature into the benchmark model. That is

collateral constraints tied to the housing values for impatient households, as in Iacoviello

and Neri (2010). This chapter tries to identify whether the Chinese housing market can be


explained better using a model with collateral constraint compared to the benchmark model

through Indirect Inference evaluation.

This chapter is structured as follows: the model with a collateral constraint is presented

in Section 4.2, and the collateral channel is also analysed. The estimation part is outlined in

Section 4.3, which introduce the data used in this chapter and the baseline calibration as well

as the estimation results. In this section, I will explore whether the model can perform well if

it introduces the collateral constraint. The standard analyses are shown in Section 4.4 that

including historical and variance decomposition and impulse response functions to different

shocks. Finally, I conclude the housing model with collateral constraint in Section 4.5.

4.2 Model

Impatient households are introduced to feature the collateral constraint. Therefore, there is

one more household on the demand side. Five types of agent exist in the economy: patient

households, impatient households, housing sector, general sector and central bank. The

key feature that distinguishes between patient and impatient households is the discount rate.

Impatient households discount future utilities more heavily than the patient ones due to the

heterogeneous preference. In each period, patient households consume, accumulate housing

and supply funds to impatient households. Impatient households work, consume, purchase

housing through borrowing from patient households. The firms in both sectors behave as

before, and I keep them as same as in the benchmark model.

Impatient Households

There is a continuum of measure 1 of impatient households. The representative impatient

households derive utility from consumption, housing purchase and disutility from supplying

labour. The representative impatient households expected discounted lifetime utility is given

4.2 Model 85

by

UIt = E0

∞

∑t=0

βIt

εpt

[C1−σc

It1−σc

+ εht

H1−σhIt

1−σh− ε

ltN1+η

t

1+η

](4.1)

where E0 is the expectation formed at period 0, β It is the subjective discount factor, the

impatient households obtain utility from consumption goods CIt , new housing HIt and get

disutility from labour supply Nt . The parameters σc, σh are the inverse of intertemporal

elasticity of substitution of consumption and housing, while η denotes the inverse of the

elasticity of work time with respect to real wage.There are three shocks in the utility function

like in Chapter 2. They are εpt , εh

t and ε lt respectively. ε

pt and ε l

t are shown here to express

intertemporal preferences shock and labour supply shock. The term εht captures shock to

housing demand 1. The impatient households’ budget constraint is

CIt + ph,t(HIt − (1−δh)HI,t−1)+(1+ rt−1)BI,t−1 = wtNt +BIt (4.2)

From equation (4.2), it can be seen that the wealth of impatient households consists of two

parts, which is shown on the right-hand side. One of the incomes comes from supplying

labour wtNt . The other comes from borrowing from patient households BIt . The left-hand

side displays the outflow of funds. The impatient households use his wealth in each period

for buying consumption goods CIt , new housing HIt with relative price ph,t and payback last

period’s debt with rt−1.

Collateral constraint

Private borrowing is subject to an endogenous limit as in Iacoviello and Neri (2010). Impa-

tient households borrow from patient households to finance their consumption and housing

purchases. The borrowers face the borrow constraint: the value they can borrow is limited by

1The detail of housing demand shock can be found in Chapter 2


a fraction of the expected present value of their housing asset. That is,

BIt ≤ mEt(ph,t+1HIt

1+ rt) (4.3)

where m is the loan to value ratio for impatient households. It should be noticed that the

change in the expected relative price of housing affects the ability of borrowing directly. This

collateral constraint can be treated as a channel to evaluate the transmission of monetary

policy shocks in the model. For simplicity, there is an assumption that has to hold when

finding the optimal behaviour of impatient households: the collateral constraint (4.3) is

always satisfied with equality. That means parameters are calibrated in a way to make it

bind. Therefore, in the steady state, the impatient households’ borrowing constraint is always

binding.

Impatient households’ Problem

The impatient households’ problem is choosing CIt ,Nt ,BIt and HIt to maximise their lifetime

utility (4.1) subject to (4.2) and (4.3). The Lagrangian:

L = E0

∞

∑t=0

βIt

εpt (

C1−σcIt

1−σc+ ε

ht

H1−σhIt

1−σh− ε

ltN1+η

t

1+η)

−λIt [CIt + ph,t(HIt − (1−δh)HI,t−1)+(1+ rt−1)BI,t−1 −wtNt −BIt ]

−λ′It [BIt −mEt(

ph,t+1HIt

1+ rt)]

(4.4)

The first order conditions:

CIt : C−σcIt ε

pt = λIt (4.5)

Nt : εpt ε

lt Nη

t = wtλIt (4.6)

4.2 Model 87

BIt : βItEtλI,t+1(1+ rt) = λIt −λ

′It (4.7)

HIt : λIt ph,t = εpt ε

ht H−σh

It +βIEt(λI,t+1 ph,t+1(1−δh))+λ

′Itm

Et ph,t+1

1+ rt(4.8)

The marginal utility losses of choosing relevant allocation are shown on the left-hand

sides of the above equations. On the contrary, marginal utility gains of choosing relevant

allocations are represented on the right-hand sides of the above equations. It should be

noticed that both the Euler equation (4.7) and the housing demand equation (4.8) are different

from the standard form due to the presence of λ ′It , which is the Lagrange multiplier on the

borrowing constraint. The multiplier λ ′It represents utility increase that would come from

borrowing, consuming in (4.7) or investing in (4.8).

Equation (4.5) links the borrower’s marginal utility of consumption to the Lagrangian

multiplier. Equation (4.6) is a standard labour supply equation which represents the sub-

stitution between labour supply and consumption. Equation (4.7) is impatient households’

borrowing. The left-hand side of equation (4.7) is the marginal utility loss, which is the

expected value of debt repayment of one unit of borrowing β ItEtλI,t+1(1+ rt). The right-

hand side shows the marginal utility gain, which is the utility gain comes from one unit of

additional consumption λIt , get rid of the loss of value due to the borrowing constraint, λ ′It .

A standard Euler condition is presented when λ ′It = 0 for all t in equation (4.7). Equation

(4.8) represents an intertemporal condition on housing demand, which requires the marginal

utility of current general goods consumption equal to the marginal gain of housing services.

The marginal utility gain of housing services shown on the right-hand sides of equation (4.8)

consists of three components: first, the direct utility from an additional unit of housing. The

second component is the expected utility coming from the possibility of expanding future


consumption relying on the realised resale value of the housing purchased in the previous

period. The third component is the marginal utility gain from the value of housing as the

collateral asset. Further imply:

Nη

t

C−σct

εlt = wt (4.9)

H−σhIt

C−σcIt

εht +β

I(1−δh)Et(ph,t+1C−σc

I,t+1

C−σcIt

εpt+1

εpt

)+(1−βI(1+rt)Et(

C−σcI,t+1

C−σcIt

εpt+1

εpt

))mEt ph,t+1

1+ rt= ph,t

(4.10)

In order to understand the behaviour of impatient households of housing purchases, I

combine the binding borrowing constraint (4.3) with the budget constraint (4.2) and obtain:

(ph,t −m

1+ rtEt ph,t+1)HIt = wtNt + ph,t(1−δh)HI,t−1 −CI,t − (1+ rt−1)BI,t−1 (4.11)

From equation (4.11), the term mEt ph,t+1/(1 + rt) represents the amount of funds

that the impatient households can borrow from the patient households. The term ph,t −

mEt ph,t+1/(1+ rt) is the instalment payment required to purchase one unit of housing. The

net worth of the impatient households in period t can be found on the right-hand side of

equation (4.11). The total worth consists of the labour income from supplying labour to

the firms plus the value of housing accumulated in the previous periods. The net worth

of the impatient households is that the total worth gets rid of the net of consumption and

debt repayment. In the long equilibrium, the impatient households use all their net worth to

finance the instalment payment that required to purchase HIt units of housing.

4.2 Model 89

Patient Households

The representative patient households derive utility from consumption and housing. In

this category of consumers, I follow Monacelli (2009) assuming that the typical patient

households are the owner of the monopolistic firms in each sector. The patient households

try to maximise lifetime utility:

Upt = E0

∞

∑t=0

βpt

εpt [

C1−σcpt

1−σc+ ε

ht

H1−σhpt

1−σh] (4.12)

where the variables in (4.12) share the similar interpretations in equation (4.1). The key

feature that distinguishes the patient and impatient households’ behaviour is the discount

factor. In equilibrium, the patient households lend to the impatient households with borrowing

constraint binding in the steady state. I assume that patient households are more patient than

impatient households, implying

βpt ≥ β

It (4.13)

The patient’s sequence of budget constraints reads:

Cpt + ph,t(Hpt − (1−δh)Hp,t−1)+St = (1+ rt−1)St−1 +π (4.14)

The right-hand side of equation (4.14) represents the total wealth of patient households,

which consists of gross returns from lending and the profits from the holding of monopolistic

competitive firms. The left-hand side of equation (4.14) is the outflow of funds of the

patient households, which contain consumption, housing purchase and lending. Following

Monacelli (2009), there are two reasons to disregard the labour supply choice by the patients’

households. First, for simplicity, I disregard for labour supply of patient households making

the level of output independent of the relative labour share of the two households. Second,

the patient households are more patient than impatient households, which prefer to hold their


wealth obtain from lending funds and from owning firms in different sectors. Hence, they

will end up owning all assets and choose to work very little in the steady state.

Patient households’ Problem

The patient households try to maximise a lifetime utility function (4.12) subject to the budget

constraint (4.14) through Lagrangian:

L =E0

∞

∑t=0

βpt

εpt (

C1−σcpt

1−σc+ε

ht

H1−σht

1−σh)−λpt [Cpt + ph,t(Hpt −(1−δh)Hp,t−1)+St −(1+rt−1)St−1−π]

(4.15)

The first order conditions:

Cpt : C−σcpt ε

pt = λpt (4.16)

St : λpt = βptEtλp,t+1(1+ rt) (4.17)

Hpt : λpt ph,t = εpt ε

ht H−σh

pt +βpEt(λp,t+1 ph,t+1(1−δh)) (4.18)

Again, in the above equations, the marginal utility losses of choosing relevant allocations

is shown on the left-hand sides; the marginal utility gains of choosing relevant allocations

is presented on the right-hand sides. For housing demand of patient households (4.18), the

marginal utility gain only depends on two components compared to the housing demand of

impatient households. That is the direct utility gain of an additional unit of housing and the

expected utility coming from the future consumption. Further imply:

C−σcpt ε

pt = β

pEtC−σcp,t+1ε

pt+1(1+ rt) (4.19)

4.2 Model 91

εht H−σh

pt =C−σcpt ph,t −β

pEt(C−σcp,t+1 ph,t+1(1−δh)

εpt+1

εpt

) (4.20)


4.3 Estimation

4.3.1 Data

The collateral constraint is introduced in the benchmark model by splitting the households

into two types (patient households and impatient households). Therefore, five more variables

involved in this model. They are consumption of patient households (Cpt), consumption of

impatient households (CIt), housing demand of patient households (Hpt), housing demand of

impatient households (HIt) and impatient households borrowing (BIt). The rest variables are

the same as that used in Chapter 3, which sample period covers from 2001Q1 to 2014Q4.

The detail of data description are outlined in the Appendix A. It should be noticed that the

unfiltered data also used in this chapter to do the evaluation and estimation.

4.3.2 Calibration

Calibrated parameter values are introduced in this section, which are divided into two groups

like in Chapter 3. The parameter values in the first group govern the dynamics of the model,

which are calibrated according to previous literature and the observations in some empirical

analyses. If the model using these calibrated values cannot pass the test, I would reestimate

these parameters using Indirect Inference estimation. The calibration values in this group

keep as same as those in Chapter 3 except for β p, β I , m. These three more parameters are

introduced since the collateral constraint is employed in this chapter. The second group of

the parameters are the steady state of the model, which are obtained from the data same in

those in Chapter 3.

In this chapter, I split out the households into two types (patient households and impatient

households) and introduce the collateral constraint between these two households. The key

feature that distinguishes between patient and impatient households is they have the different

discount factor. The discount factor of impatient households is less than the discount factor of

4.3 Estimation 93

patient households, which implies the patient households are more patient than the impatient

households. Hence, these two parameters are calibrated to evaluate and estimate. They are

the discount factor of patient households β p, the discount factor of impatient households β I .

Because of lending between two different households, one more parameter is introduced.

That is loan-to-value ratio m.

In terms of the discount factor, I calibrate β p at 0.985 and β I at 0.97 in line with Iacoviello

and Neri (2010) to guarantees that the borrowing constraint is binding for the impatient

households in equilibrium. The binding constraint captures the financial accelerator effect,

which allows the interaction between the housing sector and the rest of the economy.

m is the loan-to-value ratio, which captures the amount of loan that impatient households

can get with a given market value of the house. The maximum loan-to-value ratio in China is

0.8. However, the average loan-to-value ratio is much lower than that, around 0.3 to 0.42.

According to Liu and Ou (2017), the observed debt-to-GDP ratios of households is around

23%. Hence, I calibrate the loan-to-value ratio at 0.3, which captures the features of the

Chinese economy.

2The data are from Housing Finance Network.


Table 4.1 Calibrated Coefficients - Model with Collateral Constraint

Defination Parameter Calibration

Households:

Elasticity of substitution of normal goods consumption σc 2

Elasticity of substitution of housing goods consumption σh 1

Inverse of elasticity of labour η 0.5

Patient households’ discount factor β p 0.985

Impatient households’ discount factor β I 0.97

Loan-to-value ratio m 0.3

Firms:

Price rigidity ω 0.84

Output elasticity of capital α 0.3

Quarterly depreciation rate of housing δh 0.015

Quarterly depreciation rate of capital δk 0.03



Capital demand coefficients k15 0.5

Monetary Policy:

Taylor Rule response to inflation θπ 1.5

Taylor Rule response to output θGDP 0.125

4.3 Estimation 95

4.3.3 Empirical Results

In this section, I am going to discuss the Indirect Inference empirical results. The VARX

auxiliary model is still used in the evaluation and estimation in this chapter. The choice

of the auxiliary model is in line with the model in Chapter 3, which include total output,

housing price and nominal interest rate in the auxiliary model. I do the evaluation first. If

the model does not pass the test, the calibrated value can be predefined as starting value to

implement Indirect Inference estimation. It should be noticed that all the coefficients are

allowed to change when doing the estimation except for quarterly discount rate of patient

and impatient households (β p,β I), quarterly depreciation rate of capital and housing (δk,δh)

and the loan-to-value ratio (m) because they are identified through accounting information

or other government policy. The simulated annealing algorithm is used when applying

Indirect Inference estimation to discover the best fit set of coefficients. Table 4.2 presents

the empirical results for the model with collateral constraint. The calibration values are also

shown here for comparison.

I compare the estimated values with the calibrated values. The results show that all of

these estimated values have moved some way from the initial calibration values. On the

households side, for the estimated elasticity of substitution of consumption and housing, σc

is estimated to be 3.30 and σh has increased to 2.55, both estimated values are larger than the

initial values. The higher value of σi means that the consumption growth is less sensitive to

changes in the real interest rate. From the estimated results, the estimated σh is still lower

than the estimated σc, which implies that the substitution of housing goods is still relatively

more sensitive compared to the substitution of general consumption goods when changing in

the real interest rate. The estimated inverse of elasticity of labour supply η is significantly

larger than the starting value, which implies that labour supply inelastic. However, this high

inverse of elasticity of labour supply is much closer to other research reported by Zhang

(2009) who employed the DSGE model to study Chinese monetary policy.


On the firm side, for nominal rigidities parameters, the price stickiness ω is estimated

to be 0.33, much lower than the calibration value. The parameter ω measure the degree of

nominal rigidity. The estimated result shows that only around 33% of all firms cannot adjust

their price while the remaining 67% can adjust. This implies that the Chinese economy may

not be that sticky, which is similar to the empirical results in Liu and Ou (2017).

The value of the share of capital in production adjusts slightly, which only increase to

0.31. For capital demand coefficients, the coefficient k11 is lower than the starting value.

This implies that lower adjustment cost. The higher value of estimated k12 implies that lower

discount rate of capital. The coefficients k13 remains the same as the starting value. The

long-run relationship among coefficients in the capital demand equation is also approximately

satisfied, which is that k11 + k12 + k13 = 1.

Overall, monetary policy is estimated to be less responsive to inflation and more respon-

sive to output fluctuation. More specifically, the responsiveness of interest rates to inflation

θπ increase from 1.5 to 1.2. On the contrary, the responsiveness of output increase to 0.42

compared with the calibrated value.

This chapter aims to investigate the case when there is a collateral constraint. I want to

explore whether the model can perform well if it introduces the collateral constraint. To this

end, I compare the testing results generated from the benchmark model, where there is no

lending, with those with collateral constraint. Table 4.3 represents the comparison of the

testing results based on Indirect Inference estimation of the two models. The results show

that both models can pass on the weaker test, but the model with the collateral constraint is

obviously inferior to the benchmark model according to the Wald statistic. The benchmark

model is more probable.

In order to check both models performance with the stronger test, I add one more

endogenous variable to the existing auxiliary model. The test becomes more stringent and

powerful when I extend the features of the structural model that the auxiliary model seeks

4.3 Estimation 97

to match. The second row of Table 4.3 shows the testing results when I raise the power of

the test. The results display that the only benchmark model can pass the stronger test at 3%

significance level, while the collateral model does not pass. That implies the benchmark

model is the best model using the Wald statistic as a guide.

Different model with different coefficients may give different analysing results. Therefore,

it should be cautious when choosing the model. The benchmark model is better to be chosen

when do not focus on the lending since it has the better Wald statistic. If lending is the

important part when analysing the economy, the model with collateral constraint should be

considered. In the following, I am going to do the standard analysis such as impulse response

functions, variance and historical decomposition for the model with collateral constraint.


Table 4.2 Estimated Coefficients - Model with Collateral Constraint

Defination Parameter Calibration Estimation

Households:

Elasticity of substitution of consumption σc 2 3.30

Elasticity of substitution of housing σh 1 2.55

Inverse of elasticity of labour η 0.5 6.96

Patient households’ discount factor β p 0.985 0.985

Impatient households’ discount factor β I 0.97 0.97

Loan-to-value ratio m 0.3 0.3

Firms:

Price rigidity ω 0.84 0.33

Output elasticity of capital α 0.3 0.31

Quarterly depreciation rate of housing δh 0.015 0.015

Quarterly depreciation rate of capital δk 0.03 0.03



Capital demand coefficients k15 0.5 1.33

Monetary Policy:

Taylor Rule response to inflation θπ 1.5 1.2

Taylor Rule response to output θGDP 0.125 0.42

Trans-Wald (GDP, HP, R) 22.67 1.49

P-value 0.00 0.06

4.3 Estimation 99

Table 4.3 Comparison of the Testing Results Based on II estimation

Auxiliary Model-VARX(1) Benchmark Model Collateral Model

GDP, HP, R 1.02 1.49

(0.11) (0.06)

GDP, HP, R, C 1.92 3,46

(0.03) (0.00)

4.3.4 Indirect Inference Power Test

In the last section, the Indirect Inference testing is employed to evaluate the estimated

structural model. We would like to know how powerful the Indirect Inference test is.

Therefore, in this section, I am going to check the evaluation of the power of Indirect

Inference. In order to evaluate the power of Indirect Inference test on both benchmark

model and model with collateral constraint, I follow Le et al. (2012) and Le et al. (2016) to

conduct Monte Carlo power statistical test against parameter misspecification. Following

their research, they assume the models they used is the true model and the estimated residuals

are also treated as the true residual. Then they use the result of the Monte Carlo experiment

to establish their degree of accuracy.3 They use the Monte Carlo experiment to show that

how often the test rejects at the chosen nominal rejection rate. Their results show that

the confidence level is 5.7% when the true rejection rate at a nominal 5%. Therefore, this

experiment is fairly accurate. I employ their experiment results and treat estimated benchmark

model as well as the constraint model as the true model, using the true rejection rate at a

nominal 5% with a three variables VARX(1). Now I am interested in how the frequency of

rejection of the false model when both true models deviate increasingly from the original.

310000 Monte Carlo experiments are set up to obtain the 10000 sample of data. In each sample, theinnovations were bootstrapped to find the Wald distribution.


The false models are created by moving the parameters away from their true values (estimated

value) by x % in both directions for alternate values.

Table 4.4 displays the rejection rates at a nominal 5% given the parameter falseness from

0.5% to 7%. 4 We can see clearly from the table that when the falsity of parameters increases,

the probability of rejecting the false model increase. This implies the power is considerably

high given a significant falseness. More specifically, the benchmark model is 100% rejected

when the falsity of parameters increases to 7%. Comparing with the benchmark model,

the model with collateral constraint seems more sensitive to the increase in the degree of

falseness. The rejection rate has already increased to 100 when the degree of falseness

equal to 3. It is interesting to find that the model with collateral constraint has more power

compared with the benchmark model. It might be because the collateral model has more

restrictions, so a small change in the parameter will create the more significant overall worse

match.

Table 4.4 Monte Carlo Power test- 3 variables VARX(1)

Parameter Falseness TRUE 0.5% 1% 1.5% 3% 3.5% 5% 7%

Benchmark Model 5 7.2 9.6 13.4 41.1 63 85.6 100

Model with collateral constraint 5 20.5 57.4 88.7 100 100 100 100

The testing results in the last section show that the benchmark model passed at 3%

significance level while the model with collateral constraint did not pass when a 4-variable

VARX(1) was used. I want to check how robust the benchmark model is with respect to

misspecification and I also want to gauge to what extent the collateral model was misspecified.

Hence, this attracts me to do the Monte Carlo experiment again, but this time, the rejection

rates at a nominal 3% with a 4-variable VAR. Table 4.5 displays that how the rejection rates

vary when one more endogenous variable is included in the auxiliary models. It is interesting

4The rejection rate is obtained using true data from the true model and false model, which calculate howmany false model would be rejected by the true data from the true model with 95% confidence.

4.3 Estimation 101

to find that increasing one more endogenous variable raise the power of Indirect Inference test

as well. In this case, the benchmark model is already 100% when the falsity of parameters

just increase to 5%. Comparing with the benchmark model, the collateral model rejects 99%

of the time when the parameter falseness only raises to 3.5%. The more features that the

auxiliary model tries to match, the higher the probability that model is rejected by the data,

which is in line with the argument.

Table 4.5 Monte Carlo Power test- 4 variables VARX(1)

Parameter Falseness TRUE 0.5% 1% 1.5% 3% 3.5% 5% 7%

Benchmark Model 3 3.7 4.9 6.6 42 82.2 100 100

Model with collateral constraint 3 6.1 7.1 8.4 65.8 99 100 100

4.3.5 Error Properties on Non-Stationary Data

In the last section, I discuss the estimation and testing results of the model with collateral

constraint. Before doing the standard analysis such as IRFs, variance and historical decom-

position, I follow Le et al. (2014) to analyse the error properties. These shocks are backed

out of the model using the non-stationary data (Figure 4.1) and fit to each an AR time series

process over the period. Table 4.6 shows the stationarity of each shock and also the estimated

AR parameters. I employ two different types of stationarity test for each calculated shock

process: Augmented Dickey- Fuller (ADF) test and Kwiatkowski Phillips Schmidt Shin

(KPSS) test.

The ADF test tests the null hypothesis that the shock follows a unit root process, against

the alternative hypothesis that the shock is stationary. The ADF testing results show in

the second column of Table 4.6, we can see from the table that the labour supply shock,

housing demand shock, capital and labour demand shock in general sector as well as the

monetary policy shock reject the unit root process at 10%, 5% and 1% significant level


respectively. The p-value of the rest shocks, except for productivity shock in different sectors,

show the borderline non-rejection at 10% significance. However, the p-value of productivity

shock in both sectors approximately equals to 1, which implies a strong non-rejection of

the null hypothesis. According to DeJong et al. (1992), the very low power when errors are

autoregressively correlated is one of the problems of the ADF test. That implies the testing

does not perform well when errors are autoregression. Therefore, the KPSS stationary test is

employed to re-evaluate the structural error.

In terms of the KPSS test, on the contrary, tests the null hypothesis that the shock is

stationary against the alternative hypothesis that the shock follows a unit root process. The

KPSS testing results represent on the third column of Table 4.6. The results show that all

the shocks fail to reject the stationary except for the productivity shock in both sectors. It

should be noticed that no matter an ADF test or KPSS test, the stationarity test show that

productivity shock in different sectors contains a stochastic trend. This interesting finding

support modelling productivity shock as the non-stationary shock.

The last column of Table 4.6 shows the estimated AR coefficients of the shock process,

which allow the error data to determine it. We can see clearly from the table that many of the

estimated AR coefficients show high persistence even though those errors are stationary.

4.3 Estimation 103

Table 4.6 Stationarity of Residual and AR parameters

ShocksADF

p-valueKPSS

statistic ConclusionEstimatedCoefficient

Government spending 0.1873 0.238410 Trend Stationary 0.9292

Preference 0.3118 0.191114 Stationary 0.9048

Labour supply 0.0712* 0.242881 Trend Stationary 0.9745

TFP – General sector 0.9905 0.963698*** Nonstationary -0.6121

TFP – Housing sector 0.8919 0.932474*** Nonstationary 0.5483

Housing Demand 0.0246** 0.120453 Stationary 0.7895

Monetary policy 0.0172*** 0.052904 Stationary 0.7710

Labour demand - General 0.0647** 0.210822 Trend Stationary 0.8923

Labour demand - Housing 0.1262 0.213745 Trend Stationary 0.9893

Capital demand - General 0.0635** 0.095722 Trend Stationary 0.9017

Capital demand - Housing 0.1710 0.167694 Trend Stationary 0.9207

Notes:

1. p-value with *,** and *** rejects the unit root process at 10%, 5% and 1%

2. KPSS with ** (***) rejects stationarity at 5% (1%)


Fig. 4.1 Structure Shocks - Model with Collateral Constraint

4.4 Standard Analysis

4.4.1 Impulse Response Functions

In this section, I evaluate the role of collateral constraint by comparing the structural impulse

response functions generated from the estimated benchmark model with those from the esti-

mated model including collateral constraint. In order to make these two models comparable,

I give them the same scaled shock (0.1 standard error). In the following, I will analyse the

impulse response to housing demand shock, monetary policy shock and productivity shock in

the general sector respectively. The impulse responses of key macroeconomic variables such

as housing-related variables to traditional shocks are worth mentioning. The key macroeco-

nomic variables are total GDP, general goods consumption, housing consumption, housing

price, inflation, as well as output in different sectors. The dashed line shows the impulse

4.4 Standard Analysis 105

response of the model with collateral constraint, while the solid line represents the response

of the benchmark model.

Housing Demand shock – According to Iacoviello and Neri (2010), the housing demand

shock can be interpreted as the change in households’ preference of purchase housing. It

implies unexpected changes in individual households’ preferences caused by the social and

institutional changes or the availability of resources needed to purchase housing.

Figure 4.2 demonstrates information about the responses of housing demand shock for

the two versions of the model. We can see clearly from the figure that all variables move in

the same directions in both benchmark model and model with collateral constraint. However,

the model with collateral constraint reacts more magnitudes when there is 0.1 standard error

of housing demand shock hitting the economy. More specifically, there is an immediate

increase in the housing price and housing demand when a positive housing demand shock

occurs. It also raises the output of housing to meet the extra housing demand. The impulse

responses of selected variables to housing demand shock in both models are in line with the

main findings in many previous works of literatures.

As mentioned in the beginning, the housing demand shock can be thought as the variations

in housing demand due to the social and institutional changes. In China, the housing

institution reforms can be explained as the important institutional change. There is no

housing market before the reforms. A gradual and persistent housing institution reforms

had been launched since 1980. Hence, the demand for housing increased. The supply of

housing raised in order to meet the extra demand. Overall, the reforms in the housing market

simulated the whole economic growth, which is a significant fundamental economic change.

The more wealth the households get, the better living conditions they desire. Therefore, the

demand for housing simulates the change in the housing market and economic growth.

I also want to explore one aspect of the spillovers effect that mentioned in Iacoviello and

Neri (2010). The spillover focused on in this chapter is the relationship between housing


wealth and the general consumption. According to Iacoviello and Neri (2010), they use

the US data to study the housing market spillovers. Their empirical results show that

the collateral constraints alter the transmission mechanism, which amplifies the response

of general consumption. Hence, this propagation mechanism shows through the impulse

responses functions (IRFs) they obtained. From their IRFs of housing demand shock, the

collateral constraint presents the key feature of explaining the spillovers effect, which implies

housing demand shock lead to an increase in general consumption. However, my results

show that there is no obvious positive spillovers effect from the housing wealth to the general

consumption in China. This is maybe because the total income effect is greater than the

substitution effect according to the estimated parameters.

Fig. 4.2 Impulse Response to a 0.1 standard error Housing Demand Shock

Monetary Policy shock – Figure 4.3 shows the response of monetary policy shock. In

general, As the figure reflects, a tightening of monetary policy decreases all components of

real aggregate demand and real housing prices in both models. The positive monetary policy

shock discourages the investment and consumption in both sectors, which decrease the total


output. The decrease of housing output and housing demand shift the pressure to housing

price, which lowers the housing price as well. However,compared with the benchmark model,

this effect is not obvious on some variables in the housing sector. The lower consumption

also affects the changes in inflation.

In the benchmark model, the Taylor Rule in our model affect the consumption through

the Euler equation and therefore influence the demand for housing. As the results, it triggers

the housing price. However, in the model with collateral constraint, the interest rate also

comes in the borrowing constraint (see Equation 4.3), which add another channel to affect the

whole economy. It implies that the tightening monetary policy shock also affects impatient

households borrowing, which decreases the collateral capacity and amplifies the negative

response of consumption.

Fig. 4.3 Impulse Response to a 0.1 standard error Monetary Policy shock

General sector Productivity shock – One of the features in my model is I treat the

productivity shock in both sectors as non-stationary. In this part, I focus on the impulse

response of general sector productivity shock. Figure 4.4 shows the impulse response of


key variables to a positive productivity shock in the general sector. The non-stationary

productivity shock has the permanent effect, which influences the macroeconomic variables

including output, consumption, housing demand and these variables become more persistent

afterwards lasting over 40 quarters. In response to a 0.1 standard error technology shocks,

output in the general sector and total GDP react positively to the realisation of technological

progress. Inflation decrease due to the decreasing in marginal cost and the increasing in the

supply of general goods with a positive productivity shock.

The positive productivity shock also lowers the real housing price. we can see clearly from

the figure that the consumption in the model with collateral constraint increase less than the

consumption in the benchmark model. The reason maybe because the decreasing of housing

price drops the collateral capacity of impatient households, which implies the impatient

households borrow less. The less borrowing induces them to consume less. Therefore, the

collateral constraints alter the transmission mechanism, which amplifies the response of

general consumption.

Fig. 4.4 Impulse Response to a 0.1 standard error Productivity Shock in General Sector


4.4.2 Variance Decomposition

In this section, I focus on answering the same question as in Chapter 3 but using the

model with collateral constraint. The question is what drives fluctuations in the housing

market. Variance decomposition is also employed to investigate the contributions of shocks to

future forecast uncertainties. Table 4.7 shows the variance decomposition of GDP, inflation,

consumption and housing price at different horizons, which are computed based on the model

with collateral constraint using estimated coefficients reported in Section 4.3.3.

In terms of housing prices, we can see clearly from the Table 4.7 that the productivity

shock in general sector plays a significant role in driving the cyclical fluctuation in housing

price no matter in short run or in the long run. It explains 92.42% of the variance in housing

price in the first year and about 95% in the long run. However, the other shocks like housing

demand shock, capital and labour demand shock play a minor role in explaining the real

housing price. The housing demand shock only accounts for 1.16% of the variance in the

real housing price in the short run and decrease in the long run with 0.53% in the long run.

That is because the rapid technological change in China leads to the economic boom. The

increasing wealth leads the households in China tend to use their income to purchases houses

when the state and companies no longer to allocated urban houses treated as welfare goods.

Although the housing demand shock is the main driven force in fluctuating housing price

in previous literature, in my research, the productivity shock dominates housing price in

China. In empirically, this idea is also supported by Bian and Gete (2015). They believe

China has experienced significant development involving fast productivity progress. Higher

productivity leads to higher households’ income, which raises the higher demand for housing.

And also, Kahn (2008) argues that the productivity growth in the U.S. is a key driver of

medium to long-term movements in housing prices. In addition, the price of the new house

is also affected if the productivity growth in the construction sector is slower than in other

sectors. This idea is also approved in the case of Canada by Sharpe et al. (2001). And Moro


and Nuño (2012) also argue that this is usually the case for most countries such as Germany,

Spain, the U.K. and the U.S.

The real consumption is the non-housing variable but it does be affected by the housing

demand shock. The cyclical fluctuations in consumption are driven mainly by preference

shock and housing demand shock, which explain 39.20% and 33.70% of the variance in

housing price in the first year. In the long run, the influence of preference shock and

housing demand on housing price fluctuations decreases with around 27% and 24% in the

long run. The preference shock plays a significant role in explaining the real consumption

because the shock influences the inter-temporal Euler equation, which directly affects the

real consumption. The reason why housing demand shock also plays a role in explaining the

real consumption maybe due to, I believe, the collateral constraint. The borrowing capacity

of impatient households mainly depends on the housing price. The change in housing price

affects the borrowing, in turn, the consumption.

The following variables in the Table 4.7 GDP, inflation and interest rate are also non-

housing variables. However, the housing demand shock plays a minor role in driving the

cyclical fluctuations, which account for less than 15% of most variables. The supply shock

does affect some variables, but it is not the dominant driving force. However, the contribution

of productivity shock in the general sector to the fluctuations in non-housing variables is

more important especially for the most non-housing variable in the long run. Technology

shock explains 84% of the variances in GDP in the short run and about 91% in the long run.

In terms of inflation and interest rate, the shock explains 70%-90% in inflation and 60%-80%

in interest rate different horizons.

Technology shock in the general sector plays a significant role in the movements of the

key macroeconomic variables. These interesting findings can explain the Chinese economic

development. According to Hsu and Zhao (2009), they find that the total factor productivity

growth rates are the main reason for the economic volatility especially when China started


its market-oriented reforms. Therefore, technology becomes an engine to promote Chinese

economic growth. This is the intuition to explain why the technology productivity shock is

quite important when explaining the key macroeconomic variables.


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0.53

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0.21

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4.4.3 Historical Decomposition

I have already discussed the fluctuation of the key macroeconomic variables in terms of

variables decomposition. In this section, I focus on the historical decompositions that indicate

the contribution of each shock to the volatility of variables over a certain sample period.

Figure 4.5 and Figure 4.6 show the historical decomposition of real housing price and real

consumption. The results in historical decompositions mirror some common features with

those in variance decomposition.

According to the effects of shocks on various aspects of the economy, I classify these

11 shocks (government spending shock, preference shock, housing demand shock, labour

supply shock, productivity shock in both housing and general sectors, capital demand as

well as labour demand in both sectors and monetary policy shock) into six categories. The

preference shock and labour supply shock belong to private shocks. The public shocks contain

labour demand shock in different sectors and capital demand shock in the general sector

and housing sector. The policy shocks consist of monetary policy shock and government

spending shock.The TFP including the technology shock in two sectors. The housing demand

shock does not be classified into private shocks category since I want to observe how this

shock influence the real housing price and real consumption.

From Figure 4.5, I find that the fluctuation of housing price is mainly driven by the

technology shock and public shocks. The contribution of housing demand shock appears less

important. The figure also displays that the housing price seems relatively stable during the

period 2000-2003 and period 2010-2012, but it fluctuated a lot during the crisis period. It

should also be noticed that there was a drop started in 2007, which reflected the stock market

boom in 2007 and the market’s reaction to the global crisis. The housing market recovered

after the crisis and rise dramatically until earlier 2010. The economy increased rapidly these

years, which lead households in China earning increasing wealth. They tended to use their

wealth to purchase the house since housing market developed these years rapidly. I also


Fig. 4.5 Historical Decomposition of Real Housing Price

find that after 2010, the housing price decreased continuously as lower productivity. These

findings are generally in line with Liu and Ou (2017). They show that the housing market

triggered another significant slowdown as demand and productivity both continued to fall

from 2013 onwards, which the housing price was corrected toward its equilibrium level.

As for the macroeconomy, I focus on the dynamic of consumption. Figure 4.6 shows the

historical decomposition of real consumption. We can see clearly from the figure that the

fluctuation of real consumption is driven by private shocks like preference shock, housing

demand shock and productivity shock, which in line with the finding in variance decompo-

sition. More specifically, during period 2000-2003, the real consumption seems relatively

stable. The positive productivity shock increased the real consumption, which implies higher

productivity translated into higher household income and higher real consumption. The

negative effect can be found during period 2011-2013. The real consumption started to

fall since productivity fell. The historical shock decomposition of real consumption further

suggests the contribution of collateral constraint, which housing demand shock affected

the real consumption dramatically in late 2007. However, during period 2008-2011, the

contribution of preference shock appears more important.

4.5 Conclusion 115

Fig. 4.6 Historical Decomposition of Real Consumption

4.5 Conclusion

The aim of the chapter is that I want to explore whether the model can explain the Chinese

housing market well if I include the collateral constraint. Indirect Inference evaluation is

employed to answer this question. The testing results show that both benchmark model and

model with collateral constraint can match the data, but the model with collateral constraint

is obviously inferior to the benchmark model according to the Wald statistic. This implies

the benchmark model is the best model using the Wald statistic as a guide. Hence, it should

be quite cautioned when choosing the model. In the following, I conduct IRFs, variance and

historical decomposition to further study the model with collateral constraint. The results

show that the collateral constraint explains the spillover effect from the housing market to

the wider economy. The productivity shock in the general sector plays a significant role in

the movements of the key macroeconomic variables.

Chapter 5

Conclusion

The dramatic rise and the large fluctuation of housing price motivate me to study the Chinese

housing market. This thesis addresses two research questions related to the housing market

in China: i) the sources of fluctuations in the Chinese housing market. ii) identify whether

the model with collateral constraint enables a better performance. A DSGE model using

Indirect Inference method is employed to explore these two issues.

Following the above two motivations, I reviewed the literature about the driving forces

behind movements in the housing sector and the structured DSGE models with collateral

constraint in Chapter 2. In the existing empirical literature, the housing price fluctuation

is affected by the economic fundamentals such as construction costs, disposal income and

population. There is no consensus among researchers regarding the source of housing price

dynamics in the existing empirical literature. There are some limitations when using various

econometric models such as omitted variables problem and endogeneity problem. Therefore,

a micro-foundation structural model is chosen in this thesis to study the housing market

dynamics in China. The increasing researchers have followed Iacoviello and Neri (2010)

who use a Bayesian estimated DSGE model to discover the housing market fluctuation. More

factors are considered to enrich the model based on their analysis framework. In summary,

most of the literature that using a micro-founded DSGE model employing Bayesian estima-

118 Conclusion

tion have concluded that the housing demand shocks play an important role in explaining

the fluctuation of housing price in China. In terms of the model with collateral constraint,

I reviewed Kiyotaki and Moore (1997) who first introduced the collateral constraint and

Iacoviello (2005) who extend Kiyotaki and Moore (1997) ’s work by using housing stock as

the collateral.

Chapter 3 focus on answering the first research question: What is the driving force in

the fluctuation of housing price. A DSGE model with housing sector and some important

features of the Chinese economy is established to address this question. Two features of

the Chinese housing sector are considered in this model: i) two sectors on the supply side.

ii) the non-stationary productivity shock in both housing and general sector. In order to

check whether this model can explain the data behaviour in the Chinese housing market,

Indirect Inference evaluation is employed. The testing results show that the model using

the calibration value is rejected by the data. Hence, Indirect Inference estimation is used

to estimate the model over the period 2000-2014, which find out a set of coefficients that

can pass the test. The estimated model can fit the data well when a variety of endogenous

variables are added to the auxiliary model, explaining the output, housing price and interest

rate that I concerned about. I discovered the housing market using this right estimated model,

which can perform well in explaining the data. In terms of the driving force of fluctuations in

the Chinese housing market, the variance and shock decomposition suggest that the capital

demand shock play a significant major role in explaining the housing price. That maybe

because the housing market reform stimulates the Chinese housing industry, which develops

some new regime of capital accumulation. These regulation change on the supply side played

a key role in housing development.

The increasing interest in the DSGE housing model literature have focused on the

collateral constraint on the households’ side, which treats as a channel that connects the

housing market to the wider economy. Hence, in Chapter 4, I focus on discovering that

119

whether adding a collateral constraint to a New Keynesian DSGE model enables a better

performance. Indirect Inference evaluation is used to examine it. From the modelling point

of view, my starting point is the benchmark model that introduced in Chapter 3. In order to

examine whether the model with collateral constraint can explain the Chinese housing market

well, I include another feature into the benchmark model: collateral constraint. I add this

constraint by splitting the households into patient and impatient households. The impatient

households in the economy face a binding collateral constraint when participating in loan

and mortgage market. Indirect Inference testing results show that the model with collateral

constraint cannot provide a better performance in explaining the data. More specifically,

both benchmark model and model with collateral constraint model can match the data, but

the model with collateral constraint is obviously inferior to the benchmark model according

to the Wald statistic. This implies the benchmark model is the best model using the Wald

statistic as a guide. Hence, it should be quite cautioned when choosing the model. I also use

Monte Carlo experience to show how the power of Indirect Inference. I evaluate the power of

Indirect Inference test on both benchmark model and model with collateral constraint. The

experience results show that the power is considerably high given a significant falseness. It is

also interesting to find that the model with collateral constraint has more power compared

with the benchmark model. It might be because the collateral model has more restrictions, so

a small change in the parameter will create the more significant overall worse match.

There are two contributions in this thesis. First, the dynamic stochastic general equilib-

rium model is set up incorporating housing sector and some important features of the Chinese

economy. This provides a framework to describe the Chinese housing market in a reasonable

detail. Differing from the most previous literature, the productivity shock in both the housing

sector and general sector are assumed to be non-stationary. The non-stationary shocks could

shed light on some stylized fact in China. Second, the evaluation and estimation strategy

followed Indirect Inference method using unfiltered non-stationary data are employed in this

120 Conclusion

thesis. To my knowledge, there has been no evaluation of DSGE model with housing sector.

This is the perspective adopted in this thesis.

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Appendix A

Data

Benchmark Model

The sources of these observables from 2000Q1 to 2014Q4 are from the National Bureau

of Statistics of China (NBSC), Ministry of Human Resources and Social Security,P.R.C

(MHRSS), the People’s Bank of China (PBOC) and the Oxford Economics (OE). In this case

where the quarterly data are not available, it is only available on annual basis. I follow Liu

and Ou (2017) to convert the annual data into the quarterly data using either the ’quadratic-

match sum’ or the ’quadratic-match average’ algorithms with Eviews. The Table A.1 below

summarise all the description and sources of the data used in the evaluation and estimation.

128 Data

Table A.1 Data Description and Source of Benchmark Model

Symbol Variable Source

GDP Total Output NBSC

Yc,Yh Output in different sector Model implied2

C Total private consumption NBSC

H Total housing consumption Model implied2

W Average Wage per person MHRSS

π Quarter-on-quarter CPI inflation NBSC

N1 Total Employment Oxford Economic

Nc,Nh Employment in two sectors Model implied4

ph Real housing price NBSC

I, Ic, Ih Investment NBSC

K,Kc,Kh capital Model implied3

i Nominal interest rate POBC

Notes on Table A.1:

1. The variable Nt in the model represent the aggregate supply of labour hour of household.

The measurement of the aggregate supply of labour is the multiplication of supply of labour

in each household and the total employment. It is assumed that the working hour in the

contract is fixed (around 8 hours). Therefore, the aggregate supply of labour hour can be

equal to the total employment. In my research, due to the data limitation in China, the total

employment is used to represent Nt in the model.

2. Constructed data: Yc,Yh, H

The output in different sectors are constructed following Liu and Ou (2017). The value

of housing (Residential investment) is the multiply of the price of housing (ph) and the

quantity of housing (Yh). In this identification, the value of housing and the price housing

129

are all available from NBSC. Therefore, it is easy to the quantity of housing (Yh). The

unobservable variable Ht is obtain from the marketing clearing condition in housing sector.

The depreciation rate used in calculating Ht following Liu and Ou (2017). The output in

general sector (Yc) is calculated using the definition of GDP.

3. Constructed data: Kc,Kh,K

The total capital and the capital in different in different sectors are calculated following

Caselli(2004) using the capital accumulation equation (3.13). The investment can obtain

from NBSC. The depreciation in each sector follow Liu and Ou (2017).

4. Constructed data: Nc,Nh

Follow Barsky et al. (2007)’s assumption and the identification equation (3.44) to con-

struct Nc and Nh. Barsky et al. (2007) assume that factors flow freely across industries,

nominal wages and rental prices will be equal in each sector. That means the capital-to-

labour ratios will equalize across industries since the production function is homogeneous

of degree one no matter which sector have sticky prices and which one have flexible prices.

Consider the identification equation, there are two equations and two unknown, it is easy to

solve for Nc and Nh

130 Data


Five more variables are introduced in this model due to the inclusion of the collateral

constraint by splitting the households into two types. They are consumption of patient

households (Cpt), consumption of impatient households (CIt), housing demand of patient

households (Hpt), housing demand of impatient households (HIt) and impatient households

borrowing (BIt). The rest variables are the same as that used in Chapter 3, which sample

period covers from 2001Q1 to 2014Q4. These five variables cannot be obtained from database

directly. Hence, the observables and model equations are used to construct these variables.

The identification and steady state ratio following Andrés et al. (2013) are used to calculate

consumption and housing demand of different type of households. The unfiltered data also

used in this chapter to do the evaluation and estimation. The reason why using unfiltered data

have already been discussed in the last chapter. Table A.2 below summarise the variables

and source of the data used in this model.

131

Table A.2 Data Description and Source of Model with Collateral Constraint

Symbol Variable Source

GDP Total Output NBSC

Yc,Yh Output in different sector Model implied2

C Total private consumption NBSC

Cp Patient households consumption Model implied5

CI Impatient households consumption Model implied 5

H Total housing consumption Model implied2

Hp Patient households housing consumption Model implied6

HI Impatient households housing consumption Model implied6

BI Total borrowing of Impatient households Model implied7

W Average Wage per person MHRSS

π Quarter-on-quarter CPI inflation NBSC

N1 Total Employment Oxford Economic

Nc,Nh Employment in two sectors Model implied4

ph Real housing price NBSC

I, Ic, Ih Investment NBSC

K,Kc,Kh capital Model implied3

i Nominal interest rate POBC

132 Data

Notes on Table A.2: The detail of Note 1 to Note 4 can be found in Appendix A

benchmark model.

Note 5: Model implied data: Cp, CI . The identified equation of total consumption

goods Ct = Cpt +CIt along with the ratio of consumption of patient and impatient house-

holds following Andrés et al. (2013) are used to construct consumption of different type of

households.

Note 6: Model implied data: Hp, HI . They are calculated applying the same logic like

calculating Cp, CI . Using total housing demand together with the ratio of housing demand

of patient and impatient households by Andrés et al. (2013) to calculate housing demand of

patient and impatient households.

Note 7: Model implied data: BI . The total borrowing of impatient households can be

obtained from the binding borrowing constraint (equation 4.3). The data on the right-hand

sides of equation 4.3 are available. Hence, it is easy to get BI

Appendix B

Impulse Responses of Main Variables

Benchmark Model

Fig. B.1 Government Spending Shock

134 Impulse Responses of Main Variables

Fig. B.2 Preference Shock

Fig. B.3 Labour Supply Shock

135

Fig. B.4 Productivity Shock in Housing Sector

Fig. B.5 Labour Demand Shock in General Sector


Fig. B.6 Capital Demand Shock in General Sector

Fig. B.7 Capital Demand Shock in Housing Sector

137


Fig. B.8 Government Spending Shock

Fig. B.9 Preference Shock


Fig. B.10 Labour Supply Shock

Fig. B.11 Productivity Shock in Housing Sector

139

Fig. B.12 Labour Demand Shock in General Sector

Fig. B.13 Labour Demand Shock in Housing Sector


Fig. B.14 Capital Demand Shock in General Sector

Fig. B.15 Capital Demand Shock in Housing Sector

The Cause of Housing Market Fluctuations in China

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